Consider the $k$-sphere (I'm particularly interested in $k=3$). For each positive integer $n$ let $P_{n}$ be the set of primes that divide the order of $\pi_{i}S^{k}$ for some $i\leq n$.
Does the cardinality of $P_{n}$ tend to infinity with $n$?
Yes. In fact $\bigcup_{n\geq 1} P_n $ is the set of all primes. Serre proved that, for each odd prime $p$, there is some very predictable $p$-torsion in $\pi_{k+(p-1)}(S^k)$, for example.
There is a very nice theorem of McGibbon and Neisendorfer (proving, I think, a conjecture of Serre) that implies and vastly generalizes this.
Theorem (McGibbon & Neisendorfer). Let $X$ be a simply-connected finite complex such that $\widetilde{H}_*(X; \mathbb{Z}/p) \neq 0$. Then $\pi_n X$ contains $p$-torsion for infinitely many values of $n$.