The number of elements not in conjugate of a subgroup

For an infinite group, it is in fact possible that every element is contained in some conjugate of a proper subgroup containing just two elements. That is, it is possible for an infinite group to consist of just two conjugacy classes: the identity and everything else.

The construction of such a group is not elementary. A very nice outline of the process has been given by in Arturo Magidin's answer to Infinite group with only two conjugacy classes.

Here's an explicit example:

let $G$ be the group of increasing self-homeomorphisms of $\mathbf{R}$ with bounded support (i.e., identity outside a compact subset), and $H$ the subgroup of those increasing self-homeomorphisms with support in $[-1,1]$.

Indeed, if $g\in G$, there exists $u\in G$ such that $u(\mathrm{Supp}(g))\subset [-1,1]$, and hence $ugu^{-1}\in H$.

Also, if we restrict to those elements in $G$ (and in $H$) that are piecewise affine, for which slopes are integral powers of $2$ and breakpoints are dyadic numbers, we get a countable example for $G$.