We generally define $PH = \cup_i\Sigma_i^p$ (or various equivalent forms.) In the same notation we can also define $PSPACE = \cup_c\Sigma_{n^c}^p$--that is, like the polynomial hierarchy, but with a polynomial number of alternations between $\exists$ and $\forall$ quantifiers.
I have never seen, after looking around quite a bit, anyone define a class with more than a constant number of alternations but less than polynomial, say $\Sigma_{\log n}^p$ or even $\Sigma_{(\log n)^c}^i$. Given how many tiny differences in computational power get exhaustively studied, this confuses me. Is this a named or studied class? If not, is there a good reason why (does it reduce trivially to something else?)
While this may seem arbitrary, I think it's a natural class to consider, in particular if we look at circuit formulations: $PH$ is precisely the class of (uniform) constant-depth, exponential sized, unlimited-fanin circuits, and $PSPACE$ the class of the same circuits of polynomial depth; it seems very natural to consider logarithmic (or polylogarithmic) depth circuits.
