Many integral sequences are periodic modulo (almost) all primes. However all examples I know are either evaluations of suitable polynomials on consecutive integers (trivial examples) or grow at least exponentially fast.

Is there an interesting example with subexpontial growth (ie. the $n$-th term is eventually smaller than $(1+\epsilon)^n$ for every strictly positive $\epsilon$)?

In some sense, the interesting question is perhaps: Can an integral sequence of intermediary growth have "natural" (nice and elementary in some sense) arithmetic properties?

I used "natural arithmetic properties" in order to exclude for example the partition function. It has intermediary growth and it has definitively interesting arithmetic properties, but nobody would call them elementary, I think. (Except perhaps reincarnations of Ramanujan.)

Added: Observe that some "strong" condition is needed in order to exclude sequences of intermediate growth. $1,2!,2!,3!,3!,3!,\dots,4!,4!,\dots$ with suitably repeated terms can have intermediate growth and has only finitely many non-zero terms modulo every prime-number.