As @duskwuff suspected, I've looked into this. To answer the question, the 8085 has two extra registers in the ALU.
The 8085 has several "hidden" registers: a 16-bit WZ pair and two 8-bit ALU helper registers: ACT and TMP. WZ is part of the register file, while ACT, A (accumulator) and TMP are located in the ALU circuitry itself.
Here's a diagram of how the ALU works:
The ACT register has several important functions. First, it holds the input to the ALU. This allows the results from the ALU to be written back to the accumulator without disturbing the input, which would cause instability. Second, the ACT can hold constant values (e.g. for incrementing or decrementing, or decimal adjustment) without affecting the accumulator. Finally, the ACT allows ALU operations that don't use the accumulator.
One interesting consequence of the 8085's ALU setup is that a value can be loaded into the accumulator only after passing through the ALU.
Details on the 8085 register set are here and details of the ALU are here.