  We look at Bohemians, specifically those with population $\{-1, 0, {+1}\}$
and sometimes $\{0,1,i,-1,-i\}$. More, we specialize the matrices to be upper
Hessenberg Bohemian. From there, focusing on only those matrices whose
characteristic polynomials have maximal height allows us to explicitly identify
these polynomials and give useful bounds on their height, and conjecture an
accurate asymptotic formula. The lower bound for the maximal characteristic
height is exponential in the order of the matrix; in contrast, the height of
the matrices remains constant. We give theorems about the numbers of normal
matrices and the numbers of stable matrices in these families.

Symbolic Computation;Numerical Analysis;Numerical Analysis