Is there an algebraic number that cannot be expressed using only elementary functions? (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?
 A: I addressed this exact question in my American Mathematical Monthly paper,
What is a closed-form number? Corollary 1 in that paper states that if Schanuel's conjecture holds, then the EL numbers (i.e., the numbers expressible according to your list of rules) that are algebraic are precisely the numbers expressible using radicals.  So any algebraic number that is not expressible in radicals would be an answer to your question.  However, Schanuel's conjecture is not known to be true.  I believe that your question is still open.
A: Try the real root $\alpha$ of $z^5 + z - 3$.  This polynomial has Galois group $S_5$, so $\alpha$ is not expressible in radicals.  It is expressible using some more exotic functions such as hypergeometrics, but not AFAIK using what you're calling "elementary" functions.  
On the other hand, I'm pretty sure there is no proof that $\alpha$ is not, say, a rational multiple of $\pi + e$ (it is widely believed that $\pi + e$ is transcendental, but there is no proof even that $\pi + e$ is irrational).
