More than one time dimension
As Cumrun Vafa explains in the video linked to below the picture of him in this article, F-theory works in a total of $10+2$ dimensions. The signature of the last two infinitesimal dimensions is ambiguous, so that they can indeed both be timelike. Since these are only infinitesimal dimensions, any causality issues etc are not a problem in this case.
And as Cumrun Vafa nicely explains in his talk, F-theory gives quite a nice phenomenology with an astonishingly realistic CKM-Matrix, coupling constants, etc; so it is NOT true that theories that operate in more than one time dimension are completely off base, as some people claim. There is no reason to dogmatically dismiss every theory that has more than one time dimension.
BTW, the talk is very accessible and enjoyable.
The hyperbolicity of the associated classical field equations is lost in $d$ space plus $2$ time dimensions. One cannot define a locally SO(d,2)-invariant distinction between past an future, no matter how curled up one of the time dimensions is.
As a result, there is no way to implement causality (i.e., no way to enforce the limiting information transmission to a finite speed), and the resulting models have very little to do with the real world.
The late Irving Segal of MIT had a theory where the usual Lorentz group was replaced by SO(4,2) and there were indeed two time dimensions. His book Mathematical Cosmology and Extragalactic Astronomy, Academic Press, 1976, worked out the details. His theory has not been generally accepted, although there may be a few mathematical physicists at Montreal who are still interested in it. One of the consequences of this "chronometry" as he called it was that a part of the observed redshift was merely due to the discrepancies between the two times, and was not a Doppler effect, and thus the universe was not expanding. This theory is not currently accepted.
He was a brilliant mathematician. He understood Physics. He did not understand how to do Physics. He made some great contributions to Mathematical Physics in his theorems about operator algebras, and those theorems were motivated by Physics. In fact, he was only interested in maths that was motivated by Physics.