Do their exist power series with non circular regions of convergence? So far just about any series of the form $$ \sum_{i=0}^{\infty} \left(a_ix^i \right)$$ Has tended to have a circular disk of convergence (of some radius, sometimes even 0). Is there a reason this is always the case? Do there exist power series with say a lemniscate of convergence or an oval of convergence, etc... Or is there a clear reason why a power series must converge over a disk of some radius (possibly 0)

The Cauchy-Hadamard Theorem tells us that for every power series over $\mathbb C$, there is an $R \in [0, \infty]$ such that the series converges for $|x| < R$ and diverges for $|x| > R$. So yes, it is always a circle of convergence.

Edit: Yes, it can go both ways on the boundary. Usually that's still called a circle of convergence, tough.