Although Aristotle is not explicit about it, the answer seems to be yes. For example, Aristotle notes that the formal and final cause are often the same, and arguably multiple forms can serve as formal causes. Concerning uniqueness of the formal cause here is a relevant passage from Kenny's Aquinas on Mind (p.149):
"Whenever there is a true sentence on the pattern ‘A is F’, we can speak of the
form of F-ness; an accidental form or a substantial form as the case may be. If A is hot, there is such a thing as the hotness of A; if A is an animal, there is such a thing as the animality of A. Thus, the hotness, or heat, of a hot body is what makes it hot, and that is an example of an accidental form. The substantial form in a human being may likewise be introduced as being, truistically, that by which a man is a man, or that which makes a man a man. In each of these cases the ‘makes’ is the ‘makes’ of formal causality, as when we say that it is a
certain shape which makes a piece of metal a key..."
In other words, a form does not have to be substantial to serve as a formal cause, and accidental forms are certainly not unique for a thing. Even if we assume that only substantial forms can serve as formal/final causes it is not clear that Aristotle asserts unicity even of substantial forms. Apparently, there was a controversy about it in Aquinas's time, which suggests to me that Aristotle was none too clear on the issue:
"Now human beings grow and take nourishment, just as vegetables do; they see and taste and run and sleep just as animals do. Does this mean that they have a vegetable and animal soul as well as a human soul? Many of Aquinas’ contemporaries answered this question in the affirmative. They held that in the human being there was not just a single form, the intellectual soul, but also animal and vegetable souls; and for good measure some of them added a further form, a form which made a human being a bodily being. This was a ‘form of
corporeality’... Aquinas rejected this proliferation of substantial forms."
The "form of corporeality" comes from another well-known interpreter of Aristotle, Avicenna, see Pasnau's Form and Matter. But even if we accept Aquinas's position on unicity of the substantial form, assume (even against him) that only such a form can be a formal/final cause, and postulate some sort of analogous "ultimate" final cause in all cases, the final cause of final causes as it were, (which seems to be the idea in the OP), I do not see why Aristotle would want to restrict final causality to such a uber-cause. His four causes are devices of explanation, whenever a teleological explanation is called for in a particular context the thing will have a (context-dependent) final cause.
Think of an architect building a castle, his various construction decisions aiming at making it defensible, imposing, pleasing to the eye, etc., all of this together wraps into a final cause he is working towards. But then think of the architect's client, the king, who sees it as a strategic piece in a grand plan for expanding the kingdom, and cares little for, or does not even know, the construction details, while the architect has not a clue about the grand plan. Which is the castle's final cause? The architect's and the king's designs overlap, but none contains the other, and while we could wrap them together artificially, the explanatory value of such a chimera is dubious. There might also be a problem with this "ultimate" cause itself because Aristotle rejects actual infinity. It is one thing to potentially have a final cause in any context, it is quite another to actualize them all into an "ultimate" cause.