Is it possible to have a continuous quantized signal or a discrete infinite precision signal?
When we talk about a digital representation of a signal it is implied that we can store this representation in memory, which requires that the amount of information (bits) is finite.

An analog signal has an unbounded amount of information, both in its level and in its variation in time. To make the amount of information finite you must quantize both.

Both 'single dimensional' quantizations you suggest can be done, but each transforms an analog signal into another analog signal, that must undergo further quantization before it can practically be converted to digital.

Brain mentioned an application of discrete-time continuous-value. A LED bar that indicates the loudness of your music is an example of an (in principle) continuous-time discrete-values system.

One could argue whether a continuous signal in either dimension exists at all. Charge is quantized (number of electrons) and according to modern theories time is so too, and practical circuits have limited speed anyway.