(this is basically a repost of a question I asked at M.SE last year)

Is there an explicit real algebraic number (such that we can write its minimal polynomial and a rational isolating interval) that cannot be expressed as a combination of the constants $\pi, e,$ integers and elementary functions (rational functions, powers, logarithms, direct and inverse trigonometric functions)?

In case it is an open question, can you give an example of an algebraic number such that we do not know how to express it in elementary functions?