What are fundamental elements to existence aside time and space? Did philosopher identified elements that are necessary to existence or our existence like time and space? Is the third element matter? What might be a fourth?
Time and space are not considered fundamental, in terms of finding a theory of quantum gravity. To reconcile general relativity with quantum field theory, we know space and time must be reconciled with the field/particle picture.
The closest we have to fundamental are the fundamental physical constants, which we think parameterise our universe, but may have varied over time, and vary around the universe. For instance, the speed of light may vary in like that: our ways of measuring time are so totally bound-up with the speed of light it may have obscured these variations. The anthropic principle suggests universes where the constants aren't within quite narrow 'fine tuned' bounds, couldn't support complex chemistry, and allow minds of observers, putting us in an unlikely universe, but we could not be asking questions in more likely ones.
We know dark matter and dark energy make up 95% of what is in the universe, but we don't know what they are. We only know dark matter doesn't interact with the three quantum forces, only gravity. And that something is accelerating the expansion of the universe, which we have very few clues to explaining.
Entropy, a measure of orderliness or information, is key to explaining directionality of events (the arrow of time, ie why bulk events usually aren't reversible, but particle interactions are), like thermodynamics. At first information was thought to be a secondary property of particles. But as both relativity and quantum theories are all about what information moves where, the paradigm is increasingly that particles are secondary properties of information. The spin-networks of Rovelli's Loop Quantum Gravity, for instance.
I like the entropic gravity approach, that it is information density, or entanglement, that drives gravity (including bending time). Black holes are already thought to have the highest possible complexity down to the Plank scale; and to have the biggest gravitational fields.
There are several philosophical positions on this. The materialist position is that everything that exists is matter or states of matter. In the modern scientific form of this, that means energy, spacetime, and the rest of the ontology of modern physics.
The dualist says that in addition to matter, there is mind and states of mind, which are fundamentally different from matter and are not reducible to material things. There are various reasons for saying that mind must be distinct, such as the fact that mental objects have properties that material objects do not have. One such property is sometimes called "aboutness"; it is the property of a thought that is directed at something--a person, a place or something else. When you think about San Francisco, there is something in your thought that is directed at San Francisco in a way no material object is ever directed at anything. Another mental feature is "what it is like to be". For example, there is an experience that is what it is like to be you, but there is no experience that is what it is like to be a rock. You might try to imagine what it is like to be a rock, but the rock itself is not experiencing any such thing. Since mental objects seems to have properties that no material object has, mental objects must not be material.
Yet another alternative is idealism, which says that mind and mental states are the foundations of existence, and matter is just a perception of minds. There are various arguments for this position. Imagine you don't really have a physical body like you think you do, but are really just a brain plugged into a virtual reality such as in The Matrix. The point of this thought experiment is not that it might really be true; the points is that it is logically possible; you might have the experience of seeing, touching, and smelling a tree even though no trees exist in the universe. What this tells us is that what comes to you through your senses does not tell you if matter is real or not, because you might have exactly the same sensory experiences if it is not real. Furthermore, there are paradoxes of matter such as this one: suppose you change one of your car's tires. You still have the same car, right? Well what if you change all of the tires and all of the wheels? What if you change the engine? What if you change the doors and body? What if you gradually, over the course of a month, replace every single part of the car with a new part so not one bit of the original matter remains? Still the same car? If you say, "yes", then the car must not be a material thing because it is distinct from its matter. If you say, "no", then at what point does it become a different car? You may be identifying the car with its matter, but now it has a fuzzy identity that depends on how you look at it. Either possibility suggests that physical objects are creatures of the mind; it is the mind that organizes the external world into objects and events, those things aren't in the world on its own.
So, there are two fundamental elements of existence that different philosophers discuss: matter and mind, and depending on the philosopher either or both might exist.
There is another sort of thing that we might call abstract objects. These are things like numbers and mathematical sets and geometric figures. These aren't matter because they don't have physical properties like mass and a particular time during which they exist. They aren't mental because they don't have mental properties. Furthermore, they don't exist in time. Note that physical objects exist in time and space, mental objects exist in time but not space, and abstract objects exist in neither time nor space (or alternatively, you could say that they exist at all times and all places). Some philosophers say that abstract objects exist and other philosophers say they do not.
Another sort of thing is God, which is different from mind, matter, or abstract object. God is timeless like an abstract object, but an abstract object is causally inert, whereas God is the ultimate cause. Again, some philosophers say that God exists, others say that he does not.
There are various other possible things as well. For example, Plato taught that there is a class of things called the Forms, which are a sort of template on which material things are built. In a sense, the Forms lie on a spectrum between God and abstract objects. They are timeless and they are ultimate causes like God, but they are very restricted in what they can cause (and, of course, they aren't persons). I don't think there a many philosophers today who still believe in this. Today, the word "Platonist" generally just means someone who believes there are abstract objects.
So the conclusion is that, yes, philosophers have put a lot of effort into analyzing what are the essential sort of things that exist. This branch of philosophy is called ontology or metaphysics (although metaphysics is often taken to have a wider meaning as well). However, despite thousands of years of thinking about this, there is nothing close to consensus.
In Relativity theory the presence of mass, energy, space and time are inseparable.
- E=mc2 demonstrates the interchangeability of mass and energy.
- The Lorentz contraction and time dilation are direct effects of the interchangeability of space and time.
- Gravity is the curvature of spacetime. Not "causes", not "is caused by", it "is".
Einstein has summarised General Relativity as; "mass tells spacetime how to curve, spacetime tells mass how to move."
I have seen this holistic concept described as MEST (Mass, Energy, Space and Time) or as STEM.
All this depends of course on the immutability of physical law and the logically consistent structures which emerge from those laws. This has led some philosophers to postulate a realm of logic or order, existing independently from the physical Universe. Plato's Ideal realm can be seen as an example.
Then, there is the nature of conscious or sentient experience, what are called the qualia of consciousness. These qualities emerge neither from the laws of physics (though they parallel certain information in our heads), nor from the laws of logic (though they obey certain of these laws). They are often attributed to some realm of the Spirit.
There is much theological and ontological debate about which, if any, of these realms might be able to exist without the others, and which might not be fundamental necessities for the others to exist. Roger Penrose has proposed a kind of cyclic Eternal Triangle in which, analogous to STEM, each is dependent on one of the others while providing the fundamental foundation for the third.
If following Conifold's suggestion, you don't want to do physics, then you need to include ideas as part of existence. Then, as I see it, the problem with starting from time and space is that it is unclear whether thoughts exist in space, or space just exists in thoughts. (This, as he notes, is suggested by Kant. You can say the same thing about time, but it is less convincing. Consciousness seems to actually take up time, if it is not identical with time itself.)
You might want to take seriously Spinoza's argument that if there are basic elements of reality that base both physical and mental, there is just one. Then, since it just is what it is, it is pointless to name it. But you might as well go the Aristotelian direction and identify it with God.