Giles Gardam's paper A counterexample to the unit conjecture for group rings was a big breakthrough in group rings. Giles produces a group $G$ that is torsionfree, such that there are units in $\mathbb{F}_2[G]$ that are not of the trivial form "a nonzero constant from $\mathbb{F}_2$ times an element of $G$". He explicitly names a couple of these nontrivial units.
