Originally posted by Linden LyonsI'm not sure if this is what OP wants. Sure, in some lines you will get almost 14 possible mates from the Q/R battery, but it's only 14 because it has to be from the same position.
According to Jeremy Morse, "Chess Problems: Tasks and Records" (1995), the record for white mates is the following:
The Problemist 1946
1 h8(Q)! (threat 2 eR~), and black's various defences lead to 24 different white mates.