  We look at Bohemian matrices, specifically those with entries from $\{-1, 0,
{+1}\}$. More, we specialize the matrices to be upper Hessenberg, with
subdiagonal entries $1$. Even more, we consider Toeplitz matrices of this kind.
Many properties remain after these specializations, some of which surprised us.
Focusing on only those matrices whose characteristic polynomials have maximal
height allows us to explicitly identify these polynomials and give a lower
bound on their height. This bound is exponential in the order of the matrix.
