If you hashed a hash an infinite number of times would you end up with a unique hash?
If you repeatedly apply a generic function on its result, in a finite domain, you tend to obtain a "rho" structure: at some point, you enter a cycle whose length is (roughly) $\sqrt{N}$, where $N$ is the size of the output space for your function. In the case of MD5, $N = 2^{128}$ (MD5 outputs 128-bit values), so the cycle will have length about $2^{64}$ values. You can hash "infinitely", you will never end up on a unique value. See this answer for some details and a picture.

Well, strictly speaking, you might end on a unique value, which would be called a fixed point, but we don't know if such fixed points actually exist with MD5. For a randomly selected function, there is a probability of about $63\%$ that a fixed point exists at all. But existence of a fixed point does not mean that you will reach it. In fact it is expected that for most cases you will end in the "big inner cycle"; if there is a fixed point at all, you will ultimately hit it only with probability $1/\sqrt{N}$ or so, i.e. pretty rarely.

In all generality, when starting from some input and hashing repeatedly, you always end up walking a cycle, and that never ends. A fixed point is a special case of a cycle with only one element. For a given function, one expects that
there is a "big" cycle that you ultimately reach for most inputs, and a collection of much smaller cycles that you almost never reach.