Let $G$ be a compact Lie group and $g\in G$ be any given element in it. Consider the conjugacy class of $g$ in $G$, denoted by $[g]=\{hgh^{-1}:h\in G\}$. Our question is that:
Could you find a $g_0\in G$ such that the size of $[g_0]$, denoted by $|[g_0]|$, is infinite?
Moreover, how about the same question for compact simple Lie groups.
This size is almost always infinite. The group $G$ acts on the conjugacy class by conjugation and this is isomorphic to the coset action $G/C_G(g)$. If the conjugacy class is finite, then $C_G(g)$ is an open subgroup, which means it must be a union of connected components. So, for example, if $G$ is connected, then every element with finite conjugacy class is central. If $G$ is simple, then there are only finitely many such elements.
Consider a case like $GL_n(\mathbb{C})$; the conjugacy classes are elements with the same Jordan normal form. There are only finitely many elements with a given Jordan normal form if it's a scalar matrix $\lambda I$.