I would think of it as a limit of the process
ΩS2 = ΩΣS1
Ω2S3 = Ω2Σ2S1
Ω3S4 = Ω3Σ3S1
ΩnSn+1 = ΩnΣnS1
so that πk(St) would be Z, Z2, Z2, Z24, 0, 0, Z2, etc.
Er, no, they plainly don't. ΩΣS0 is Z, but ΣΩS0 is trivial.
Furthermore, I see from cracking open my copy of Hatcher's "Algebraic Topology" that of course you can cobble together any space with any homotopy groups you like (as long as the ones at n>1 are abelian) from K(G,n) spaces. So yeah, the renormalized S∞ exists, but I dunno whether it's a monstrous gross blob that nobody studies, or whether it arises naturally somehow.