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)
 A: 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.
A: Every power series has a well-defined radius of convergence. If this radius is not $0$ or $\infty$, it defines a circle in the complex plane. Inside this circle, the series converges; outside the cirlce, it diverges. On the circle itself, it can converge or diverge.
The reason, briefly, is this: if the series $\sum_{i=1}^\infty a_iz^i$ converges, then the sequence $a_iz^i$ is bounded, i.e. $|a_iz^i|\le M$ for some $M \in \mathbb R$. So if $|w| < |z|$, then $\sum_{i=1}^\infty a_iw^i$ converges because each term is at most $M\left(\dfrac{|w|}{|z|}\right)^i$ in absolute value.
A: Power series always converge on a disk. It comes from the fact that:
For every $z_{0}\in\textbf{C}$ such that $(a_{i}z_{0}^{i})_{i\in\textbf{N}}$ is bounded, then we have for every $z\in\textbf{C}$:
$|z|<|z_{0}| \rightarrow \sum_{i\geq 0} a_{i}z^{i}$ is absolutely convergent.
In other words, there is a radius (the supremum of the $|z_{0}|$s) such that if the absolute value of $z$ is strictly inferior to the radius, the series is convergent, and if said absolute value is strictly superior to the radius, the sequence which defines the series is not even bounded, and therefore the series is not convergent.. This defines an open disk of convergence. There can be isolated points of convergence on the frontier of the disk, though, but nothing more exotic. 
