Suppose you have an DGalgebra $A$, and a DGbimodule $M$ over $A$. Under which conditions will the rank of the bimodules $H_*(M^{\otimes_A n})$ will grow exponentially in terms of $n$? Here $M^{\otimes_A n}$ is the Abimodule $M\otimes_A M\otimes_A\cdots\otimes_A M$ (with $n$ terms). Anything, even a counterexample would be interesting..
I don't know in general, but in Heegaard Floer theory, there are bimodules naturally associated to a mapping class of a surface selfhomeomorphism. The rank of $H_*(M^{\otimes_A n})$ grows exponentially iff the underlying mapping class group element is pseudoAnosov. See our paper at http://front.math.ucdavis.edu/1012.1032. In particular, it's easy to give examples where the growth rate is linear, although I'm sure there are also more elementary constructions. (There are also other, earlier constructions for the braid group, by KhovanovSeidel and KhovanovThomas. I don't know if they explicitly stated the fact about growth rates, but it follows directly from their results.) 

