There are infinite families in the stable homotopy groups of spheres; many of these can be seen by looking for "periodicity" phenomena in the Adams-Novikov spectral sequence. An example is the image of the $J$-homomorphism. This particular infinite family can be described algebraically (at least at odd primes). If I understand correctly, these elements correspond to elements of filtration $1$ in the ANSS. That is, they are in $\mathrm{Ext}^1_{BP_*BP}(BP, BP)$ via the exact sequence of comodules $$0 \to BP \stackrel{p}{\to} BP \to BP/p \to 0.$$ The invariant elements in $BP/p$ are the powers of $v_1$, and the coboundaries of these live in $\mathrm{Ext}^1(BP, BP)$. These correspond to multiples of the image of $J$.

Other infinite families in the stable homotopy groups of spheres can be obtained from a similar procedure. One nice thing about the image of $J$, though, is that it has a geometric interpretation: if the stable homotopy groups of spheres are identified with the framed cobordism groups, then the image of $J$ corresponds to framed spheres. Are there similar geometric interpretations of these more exotic infinite families?