A personal list.

• Hope the hypothetical author would not restrict the base to be a field, even perhaps allowing another DGA as base. (Most of the following comments probably assume this.)
• The flatness hypotheses over the base and Shukla homology as a repair for this. Invariance of Shukla homology under weak equivalences of DGAs.
• Relationship to Tor^R⊗Rop(R,M).
• For R→S a map of commutative DGAs with S flat over R, the Hochschild homology spectral sequence starting with HH^R(S, Tor^R(M,N)) and converging to Tor^S(M,N).
• Base-change formulas, ranging from simple ones to the following: $$ S \otimes_{HH^R(S)}^{\mathbb L} HH^R(A,M) \simeq HH^S(A,M) $$