For a natural number $k$ (0 is a natural number), let $T_k$ be the collection of all languages that can be efficiently decided by quantum circuits consisting of Clifford gates and at most $k$ $T$-gates (that is, we modify the definition of $BQP$ so all circuits consist only of Clifford gates and at most $k$ $T$-gates).
Is it possible to show that $T_k \subsetneq T_{k+1}$ for each $k$? In particular, can we show that $T_0 \subsetneq T_1$?
I think your hierarchy collapses, or at least would never get beyond $P$, following the top-line results of Bravyi and Gosset.
Bravyi and Gosset's paper gives an algorithm to classically simulate a quantum circuit on $n$ qubits comprising $O(\mathrm{poly\:}n)$ Clifford gates and a constant number of $T$ gates - that is, polynomial in $n$ (although exponential in $T$).
You might have a hierarchy within $P$, but at any level of your hierarchy, you're not dependent on $n$.
