[math-fun] Conjecture on Skewes' number

Skewes' number, e^e^e^79, is a notoriously flamboyant upper bound in analytic number theory. As an upper bound, it has since been replaced by much more moderate results. A <a href=" http://garden.irmacs.sfu.ca/?q=op/is_skewes_number_e_e_e_79_an_integer">conjecture</a> on the Open Problem Garden is that Skewes' number is not integral. I am boggled. Of course we know that e^79 is nonintegral because e is not algebraic. (Is there an easier proof?) But do we even know that e^e^79 is nonintegral? Do we know that e^e^10 is nonintegral?