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A little reading suggests:

It is known that either $\pi + e$ or $\pi \times e$ is transcendental (or possibly both), but no proof is known that one of those two numbers in particular is transcendental.

If we just want irrationality rather than transcendence, is a proof known?

Can we prove $\pi+e$ is irrational? Can we prove $\pi \times e$ is irrational?

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It is not known whether $\pi + e$ is irrational, nor whether $\pi \times e$ is irrational. See $\# 22$ here.

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  • $\begingroup$ I typeset your answer to change * to $\times$ and to add the webpage as a link. Hope you don't mind. $\endgroup$
    – user17762
    Commented Mar 21, 2011 at 4:51
  • $\begingroup$ @Sivaram: Not at all, thanks for cleaning it up. $\endgroup$
    – Alex Becker
    Commented Mar 21, 2011 at 4:53
  • $\begingroup$ The link does not work anymore. Here is the version from Google cache. $\endgroup$
    – Martin Sleziak
    Commented Jun 17, 2012 at 6:56

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