Are there pseudorandom number generators (PRNG) with no finite period?
If the state of the PRNG is finite, then it has a finite period.  (By finite, I mean the same as we mean when we say that a finite-state automaton is finite: the set of all possible states is finite.  For instance, if the state always fits into $b$ bits, for some fixed value of $b$, then its state is finite.)

In practice, worrying about the period of the PRNG may be akin to arguing over angels on a pinhead.  A 256-bit state is large enough that a well-designed PRNG will never repeat within the lifetime of the universe.  Therefore, concerns about repetition are pretty much irrelevant for any reasonable, well-designed PRNG.

In practice, the real challenge is to make the PRNG unpredictable (or close enough for the application's purposes); ensuring it won't repeat is much easier, relatively speaking.