Is there something such as an infinite potentiometer?

The device Spehro Pefhany's answer builds out of two pots is actually available as a single unit, for example the ALPS RDC803101A.

If you turn it, you get two sawtooth signals that are 180° out of phase, so when one output is in its “dead zone”, you can use the other one instead to determine the position.

This model also has no detents, so no clickyness at all.

The best continuous rotation sensor I've used is the AMS series.

Something like this might suit your purpose.

Of coursed they can't support infinite output values, but associated with an MCU you can set it to mid scale each time you turn your unit on, or remember last settings.

There are also plenty of relatively cheap optical encoders that would allow the same strategy of infinite rotation. For example, this from Bourns

I have not seen something like that, does not mean that it does not exist. It does not particularly lend itself to construction with normal pot designs, including modular ganged pots, because the leads usually come out of one side.

There are pots without end stops, however they have a dead angle. Expensive long-life ones are called 'servo pots'.

You could make such a device by mounting two of these pots on opposite sides of the board (phased differently, obviously).