An example of a unipotent matrix which is NOT upper triangular Define::=  $I_n$ -- the $n \times n$ identity matrix.
Let $A$ be an $n \times n$ real matrix.
Define::=  Nilpotent matrix -- an $n \times n$ real matrix $X$ such that $X^n = $ the zero matrix for some $n$ in the positive integers.
Define::= unipotent matrix $U$  -- $A - I$ which is nilpotent.
I'd like an example of a unipotent matrix which is not upper triangular

What you're asking is whether there can be a non-triangular nilpotent matrix. For example, consider a matrix $A = I_n + uv'$, where $u,v\in(\mathbb{K}\setminus\{0\})^n$ such that $\langle{u,v}\rangle = 0$. Since $u,v \neq 0$, $uv'$ and by extension $A$ are not necessarily triangle matrices.
Furthermore, $A$ is unipotent due to: $$(A - I_n)^n = (uv')^n = u(v'u)^{n-1}v' = 0$$
