With regards to some of the comments towards the bottom from David Ben-Zvi and Tim Perutz: you can get around some finite dimensional restrictions. Specifically, there is an interpretation of the Hochschild cohomology of $C^*(BG)$ in terms of string topology. Namely, it's an inverse limit of the homology of a pro-object that approximates the free loop space of $BG$ by finite dimensional manifolds. As hinted at by Dan Ramras' comments, a lot of this comes from Kate Gruher's work. This comment is explained in detail in a paper of hers and mine: arXiv:0710.1445.

With regards to some of the comments towards the bottom from David Ben-Zvi and Tim Perutz: you can get around some finite dimensional restrictions. Specifically, there is an interpretation of the Hochschild cohomology of $C^*(BG)$ in terms of string topology. Namely, it's an inverse limit of the homology of a pro-object that approximates the free loop space of $BG$ by finite dimensional manifolds. As hinted at by Dan Ramras' comments, a lot of this comes from Kate Gruher's work. This comment is explained in detail in a paper of hers and mine:

• String topology prospectra and Hochschild cohomology, Journal of Topology, Volume 1, Issue 4, October 2008, Pages 837–856, https://doi.org/10.1112/jtopol/jtn025 arXiv:0710.1445.