How many levels of pointers can we have?

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The C standard specifies the lower limit:

5.2.4.1 Translation limits

276 The implementation shall be able to translate and execute at least one program that contains at least one instance of every one of the following limits: […]

279 — 12 pointer, array, and function declarators (in any combinations) modifying an
arithmetic, structure, union, or void type in a declaration

The upper limit is implementation specific.

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