We must first distinguish between what is physically possible — what it is possible to actually occur — and what is imaginable, or logically possible under certain premisses.
Remarks about the logically possible
Logic itself — which I will take to mean classical propositional logic — has very little to say about time, or about infinite chains of consequences, either extending forwards or backwards. Indeed, it has nothing at all to say. Logic is merely a tool which we use to investigate topics, but anything it has to say on the subject are from premisses which we supply. So what is logically possible depends on the premisses we adopt.
Obviously, if we assume that there cannot be infinite regresses, we will conclude that infinite regresses are impossible; and if we assume that everything must have a cause, then infinite regress is necessary. Boldly asserting our assumptions is not a form of logical deduction, however. So we must try to avoid doing so if we wish to consider logical possibility or necessity.
We can observe that the two statements — everything must have a cause, and that there cannot be an infinite regress of causes — are in apparent conflict with one another. There is one possible resolution: a cycle of causes, where A ⇒ B ⇒ C ⇒ A, and the like, including potentially complicated networks of mutual-causation. If you find this just as dissatisfying as an infinite regress of causes or an uncaused event, then you may which to assume that such cycles cannot exist: but then you should remain aware that this is an assumption on your part.
There is absolutely no proposition A that we know of, which "causes" another proposition B to hold — that is, where A ⇒ B — which prevents us from considering yet another proposition Z such that Z ⇒ A, and where we may regard A as true because Z is true. So every proposition can be concieved of as being caused by another. But there is nothing which forces us to formulate such a proposition Z, either. We must move beyond mere sentential logic if we wish to plumb this idea further.
Infinite regresses in mathematics
We may consider what ideas come from mathematics to inform our ideas about whether logical causal chains are possible: mathematics is in effect our most intense testing grounds for logical consistency of ideas. Indeed, in modern mathematics, infinite forward-moving causal chains are common. The simplest example is Mathematical Induction, in which one proves that if some property P holds for 0, and if P(0) ⇒ P(1), and if P(1) ⇒ P(2), and so forth ad infinitum, then P holds of all whole numbers: one essentially completes an infinite chain of implications in one swoop. It is similarly common to build "upward towers" of containments: for example, sets A ∈ B ∈ C ∈ D ∈ ... However, it is unusual to consider chains of conditionals which reach "infinitely backwards", where ... ⇒ Q(3) ⇒ Q(2) ⇒ Q(1) ⇒ Q(0); and in most formulations of set theory, chains of the form ...∈ D ∈ C ∈ B ∈ A are expressly forbidden. We must not mistake this for logical impossibility, however. The axioms of set-theory that we have today were explicitly formulated to avoid confusions of definitions of sets, but they are not the only such formulation: there is a study of so-called non-well-founded set theories in which such "infinitely descending chains" are possible. As to infinite chains of consequences, for any predicate P for which we have an infinite ascending chain P(0) ⇒ P(1) ⇒ P(2) ⇒... of entailments, the predicate Q(n) ≡ ¬P(n) has an infinite descending chain ... ⇒ Q(2) ⇒ Q(1) ⇒ Q(0) by contraposition. So if you admit infinite chains of "logical effects", you must also allow infinite chains of "logical causes" as well, or very carefully re-examine your foundations of logic.
Infinite descent is very common in mathematics, of course, if you consider the set of the integers ... < -3 < -2 < -1 < 0 < 1 < 2 < 3 < ..., or similarly if you consider the rational or real numbers ... < 1/16 < 1/8 < 1/4 < 1/2 < 1. Arguing for the fact that these are defined in terms of the whole numbers starting from 0 neglects the fact that we have chosen that starting point for reasons which may be described as simply traditional; the fact that we feel compelled to consider number systems which allow these infinite backward regresses is also a counterpoint.
So much, then, for inspiration through mathematics.
What are the permissible assumptions to use?
It is difficult to see how to proceed without entering the domain of physics (which I will touch on momentarily).
The Principle of Sufficient Cause is very much in sympathy with determinism; but of course assuming that the world is deterministic does not prevent us from entertaining the idea of a further cause to any particular cause that we might like to imagine — so more physical assumptions beyond mere determinism would be necessary to make the notion of determinism useful.
Consider an "actual infinity" of regress — that is, where one may not only posit a preceding cause for any cause, but actually entertain a completed chain of causes. One might try to argue that an actual infinity of anything (logical causes or otherwise) is absurd; and while this was an active debate in philosophy of mathematics in the late 19th and early 20th centuries, the consensus is heavily in favour of actual infinities; a simple rejection of actual infinities is not likely to be convincing to others. But even if you only admit potential infinities of causes, you still have a potentially-infinite-regress, where the only reason why you don't entertain a cause for some early event is because you get caught up in doing something else instead. (The tendancy for us to do so is a possible reason why the idea of a first cause is so popular in the first place.)
The fact that there may, or may not be, a largest infinity which describes an infinite regress, is more ambivalent. Few people are terribly concerned about the subject so far as I can tell. However, the fact that one could always posit "a larger infinity", a la Cantor, is no rebuttal against an actual infinity of causes (despite the fact that this is in effect what Aristotle does for his Prime Mover): there is also nothing preventing someone from positing a cause for what otherwise would appear to be a Prime Mover. Whether one prefers a system of reason in which largest possible cardinalities exist, or do not, is a matter of taste; this is an impasse for the debate.
If you are of a religious persuasion — and in particular, a creationist — then it will seem quite natural to posit that there is a first cause. Suffice it to say that there are many people who will find your arguments unconvincing, if for no other reason than the fact that they do not agree to the assumptions included in your religious background.
I am unaware of any particularly compelling ideas — or for that matter, any particularly interesting ideas — which would decide in favour either of infinite regress, or in favour of the impossibility of infinite regress, as logical necessities. As far as I can tell, both the notion of a first cause and the notion of an infinite causal regress are logically coherent — except if you in essence assume that one of them is false.
It would seem that there is nothing left but to get out of the arm-chair, so to speak, and actually look at the outside world to see what is more likely to be the case.
Remarks about the physically possible
Because this is not the Physics StackExchange forum, we should not pretend to answer definitively what is "actually" possible, which is the domain of physics (or science more generally). However, we may make some observations from what is broadly known in the physics community.
For physical quantities or qualities, such as mass and energy, physicists tend to be skeptical of the idea of infinite quantities, if for no other reason than the fact that some object of infinite magnitude should presumably be easy to spot (if it didn't destroy the universe first). However, things like "age" aren't physical quantities or qualities; the universe may have processes which we can use as reliable time-keeping devices, but time is not written into matter itself, so far as we know.
Remarks on cosmology
Of course, the Big Bang theory is a physical theory, and it posits that our universe has a beginning only a finite amount of time ago. So this supports the idea that the universe does not have infinite causal regresses. But this is an observation, not a theoretical proof: our universe happens to be finitely old, and only so far as we can tell. (As if we could do better than the best of our observations.) When Einstein formulated General Relativity, he postulated a cosmological constant precisely because he thought the universe was in an infinitely old steady state: this is a move he later described as the biggest mistake of his life, but only because his prejudices prevented him from making one of the most astonishing anticipations in the history of science — modern theoretical grounds for a finite age to the universe based on a theory of gravity (which would have been an unanticipated event on the order of magnitude as Dirac's prediction of antimatter). These days, people feel more forgiving of Einstein's mistake, because it would seem that there is a non-zero cosmological constant, just as Einstein thought — only it has the opposite sign to what he thought, so that the universe is not only expanding, but faster than it had before.
The continuity of time
Of course, if time is continuous, there actually are infinite causal chains, but more of a Zeno-like flavour: between every cause and effect which happen at different times, there are intermediate effects and causes, and ones between those, ad infinitum. In the limit of infinite division of causal chains, you can obtain a continuum of intermediate events. Alternatively, between a cause at time t=0 and at time t=1 you may contemplate intermediate events at t=1/2, t=3/4, and so on for all times t=1-2-n for all positive integers n, still giving rise to an infinite chain of events which lead up to the event at time t=1. This is only prevented if there is discrete time; but there's no particular evidence that time is finite. (There is indeed research into such discrete models of time, and although this research sometimes looks interesting and promising, there isn't anything particularly strongly suggestive.)
On determinism and causation
It is possible that we might be able to undermine the Principle of Sufficient Reason, if for instance there are random events. Do they have causes, and if not, can the whole universe (or some powerful entity within it) perhaps be uncaused? Of course, many events which seem random can in principle be predicted if we have enough information about the initial conditions in which the die was rolled. Chaos theory may predict that it is impossible to pin down initial conditions sufficient to predict for all subsequent times, and quantum mechanics suggests that perhaps there is no perfectly defined initial conditions in the way that we would require e.g. in Newtonian mechanics. But so persuasive is the idea that the world acts according to deterministic and causal mechanisms that it is difficult to abandon the idea that everything happens for a reason, and so there are researchers such as those who work on de Broglie-Bohm theory who seek to give a deterministic interpolation of quantum mechanics (as opposed to "an interpretation", speaking here against the common manner of speaking of de Broglie-Bohm theory).
Of course, here too there is an impasse: just because one of our more practical theories can be formulated efficiently as a probabilistic theory, does that mean that therefore there is randomness inherent to nature? But this is just a particular instance of the problems of epistemology: if nature is sufficiently subtle, it can fool us into classifying it differently than we might if we were somehow more perceptive or less biased. As with everything in science, the jury is still out.
But the state of the art in theoretical physics is that there's no theoretical grounds for ruling out infinite causal chains, even ones which extend into the infinite past: the best we can do is to say that observation suggests.
This is ultimately what matters anyway; what is, rather than what might otherwise be.
What observation — and our interpretation of these observations — has thus far suggested that our universe is finitely old, and that it is reasonable to suppose that there are events which are not completely characterized by what came before. However, by the very fact that we have a useable theory of random behaviour in quantum mechanics, no event seems to be "completely uncaused"; and it also assumes a continuous evolution in time, so that there are at least Zeno-like infinite chains of cause and effect.