> Analog signal processing is clearly less memory than a register, no?

You are going to have a hard time doing analog signal processing with memoryless elements. In the linear domain all you can do is apply gain and mix signals together. If you work with memoryless nonlinearities you can do waveshaping, which is generally only useful when applied to special signals (e.g. sine waves).

Any time you want to do frequency-dependent behavior (filtering, oscillation) you need energy storing elements, usually capacitors, sometimes inductors. A capacitor is just like a register: it stores charge, similarly, inductors store energy in the magnetic field. Needless to say these devices are not memoryless. In fact, since the quantity that they remember is a continuous variable, they store a lot of information.

> Analog signal processing is clearly less memory than a register, no?

Bucket-brigade delay lines?