How can one prove the existence of potential energy? Potential energy always seemed weird to me. Like it's not a real type of energy, just something made up so that the Law of Conservation of Energy stays true. So I want to know if the existence of potential energy can be proven without the use of conservation of energy. And if that is the only way to prove it, then how can conservation of energy be proven without potential energy? Basically I'm looking for how this circular argument, which is the only I've ever gotten, is justified.
 A: Take a rubber band, stretch it, and leave it stretched for a day. You have just stored energy. This rubber band now has potential energy, energy that could potentially be released and converted to other, more apparent types of energy. 
Come back in a day, you will likely find that energy still stored. Cut the rubber band quickly and watch it jump around at it releases that potential energy into kinetic and heat.
There are other ways to store/create potential energy. Suspend an anvil from the ceiling. Come back in a day. Cut the chain. Watch the potential energy converted to motion and probably break something. It took you a lot of work to get that anvil on to the ceiling, and all of that energy was released upon cutting the chain. You gave it the potential to come slamming down to the ground by hanging it above (or away from) a large body with gravitational pull (the Earth).  Similar to the rubber band.
The conservation of energy thing helps you understand that the work you put in to hanging the anvil or stretching the rubber band will not disappear. It is still there, waiting. 
Without using conservation of energy in the proof
But you can still see potential energy without addressing conservation of energy. Happen upon a stretched rubber band or a suspended anvil. If there is something you can do which releases energy, then the system had potential energy. Gasoline, natural gas, balloon filled with air also examples. One little match or poke and energy is released. 
If you “poke” the system and nothing happens, there is either no potential or you haven’t poked the right way. 
If you have to poke it, the energy is “potential” by definition. It had the capability to release energy. If it didn’t require a poke, it would be kinetic or other forms of energy. 
A: Let $\mathbf{F}(x,y,z)$ be a vector field: if a function $V(x,y,z)$ exists such that the above vector field can be expressed as $\mathbf{F}(x,y,z) = -\textrm{grad} V(x,y,z)$ in any point of its domain, then the function $V(x,y,z)$ is defined as the potential energy associated to the vector field $\mathbf{F}(x,y,z)$, that in turn is said to be conservative.

So I want to know if the existence of potential energy can be proven without the use of conservation of energy

As you can see from the above definition, conservation of energy is by no means invoked in the definition of potential energy.
Whenever a field is conservative in a simply connected domain, it can be shown that the work done by the field along any path does not depend on the form of the path, rather it only depends on its extrema. As a consequence, the work done by a conservative field on a closed curve vanishes.

Let us now consider a point particle subject to external forces, obeying the Newton's equations; let moreover $T(v_x,v_y,v_z) = \frac{1}{2}m |v|^2$, a function of the velocity of the particle that we refer to as kinetic energy. It can be shown that the work done by a point particle along any path can be written as the difference of the above function in the extrema of the path, namely
$$
W_{\gamma}(A\to B) = T(x=B) - T(x =A). 
$$
However, in general, the work can be always decomposed as the sum of the work performed by the conservative forces plus the work performed by the non conservative forces: $W = W_{\textrm{cons}} + W_{\textrm{non-cons}}$; plugging the above in one obtains:
$$
T(B)- T(A) = W_{\gamma}(A\to B) = W_{\textrm{cons}} + W_{\textrm{non-cons}} = V(A)-V(B) + W_{\textrm{non-cons}}
$$
where we have used the fact that, if any non-conservative force exists, then it has to be generated, by definition, by its own potential energy. The above becomes:
$$
W_{\textrm{non-cons}} = (T(B) + V(B)) - (T(A) + V(A)) = E(B) - E(A)
$$
where we have defined the total energy of the particle in any point $(x,y,z)$ as the sum of its kinetic term plus the potential energy of the field calculated in that point. As such, one obtains that the work done by the non-conservative forces equals the difference of the total energy calculated in the extrema of the path. If no non-conservative forces come into play, then the left hand side vanishes and so does the right hand side; we therefore say that in those cases the energy is conserved, as its value in $A$ must equal its value in $B$, $A, B$ being *two any points$.


Like it's not a real type of energy

There is no such thing as "real energy". Energy is defined as a function of the coordinates $(x,y,z)$ as $T(v_x,v_y, v_z) + V(x,y,z)$ and it is, thus, simply a function and no more. It can be related, though, to the work done by the non-conservative forces, as seen above.
A: To offer an akternative view: (electromagnetic) field theory actually disagrees with the view of potential energy. Take two charges, if you write down the potential energy you have V=k qQ/r. But this is the same (neglecting self energy) as $\int E^2 dxdydz $. So if you say that the latter gives the energy of the electric field, you dont (cant actually since you should not count the same term double) use the potential energy for energy conservation.
A: 
just something made up so that the Law of Conservation of Energy stays true. 

Physics is about observations and data fitted with mathematical models. The mathematical models are general, controlled by mathematical axioms which test the truth of mathematical statements. For the mathematics to apply to the physical observations one needs extra axioms, called laws, principles, postulates , which choose a subset of the mathematics that can fit observations and data and, very important, predict values for new setups.
One of the most general laws is the law of conservation of energy. It has been postulated so that the measurements can be fitted with a simple mathematical model. Of course one enters circular arguments when trying to prove "laws". Laws cannot be proven, in the same way that mathematical axioms cannot be proven. They are postulated.
Take a lake up in the mountain. The water running down can be used to extract a lot of energy, light up a city.  The mathematical model used calls potential energy the energy that can be extracted once the water reaches a hydroelectric plant. The model of gravity is consistent with calculating the energy needed in the gravitational field for the water to be transferred to the lake from the surface, to give the potential for getting energy when it flows back down. That is why it is called potential energy. Conservation of energy is the law to  nail this observation to the theory of mechanics.
A: Potential energy is real but also weird as the question suspects. It is real because we can feel it. As given in the other answers, a clamped compressed spring jumps on the break of the clamp, and the fall of an absolutely static rock from a mountain can do a lot of damage(work). The explosion of a bomb that stayed dormant for years underground is another example. Energy is defined as the ability do work, and it is clear that we are having a dormant energy in all these cases- electrical in the first and third examples(chemical) and gravitational in the second. Experiments shows that energy is strictly conserved. The work done in compressing the spring or raising the rock or creating the chemicals for the bomb can all be recovered in full if we neglected losses. They should really be called leaks not losses, as these looses will appear as energy elsewhere.
But where is this potential energy stored and how can it stay dormant for so long. This is the mystery we are trying to clear. The mystery becomes even deeper if we recall that matter is always on the run, and the smaller matter particles become the faster they run- till reaching their limit near the speed of light around the electron mass. So clearly there is nothing dormant anywhere for the potential energy to hide in!
But we can shed some light on it if we do the following; imagine a steel ball rotating within a circular strap placed on a smooth table. The ball exerts a force on the strap due to the centrifugal force, or to be precise due to the ball changing its direction of motion(momentum) continuously(with magnitude fixed). This ball system behaves very much like our compressed spring example. If we somehow pushed and reduced the diameter of the strap circle slowly, We do work, the ball speed increase and the centrifugal force increases too. The work done on the ball is the force times distance moved towards the center. If we now allow the strap to expand instead and become larger, the speed goes down as the ball do work this time. Using little algebra we can say(with approximation); 
Pot.Energy= Integral(force x distance) =∫F.r dr=∫ m.a dr =m ∫v^2/r dr
=m ∫(ωr)^2/r dr=m ∫ω^2 r dr)=.5m ω^2 r^2=.5m(ω r)^2=.5mv^2 = change in Kinetic energy! F=mv^2/r is the centrifugal force, and v=ωr with ω constant.
So we come to the conclusion that potential energy resides in the spring force resulting from an enforced path-curvature of moving particles. To put it in more terms we say; ''Energy is stored as kinetic energy when particles move linearly, and stored as potential energy when particles move on a curved path. Clearly particles can't move on a curved path on their own- the need to be pushed into it. Hence a potential energy store must involve many particles. A large number of particles in a small space have a large aggregate inertia and can take the role of the strap on the compressed spring in the example above. That is why kinetic energy can refer to just a single particle, whereas potential energy need to be defined within group of particles that can force each other to go in smaller or larger circulation paths. 
It is useful to also note that energy is conserved because momentum is conserved- since we can write; E=(1/m)∫mv.dmv=.5 mv^2 for constant m, p=mv is the momentum. If we assume the particles of a gas are doing an effective rotation(but appearing as random motion), we can even derive the equation of state for a gas with the pressure obtained from the centrifugal force per unit area. This demonstrates clearly that the seat of pressure(potential energy) is in the forces resulting from all the aggregate change in direction of the moving mass. This also makes it clear why the virial theorem can connect the potential and kinetic energy (sum of kinetic energy is half the sum of potential energy) in any group of similar interacting particles under a conservative force. 
