What's the loop space of a circle?

$\Omega S^1\cong \mathbb Z$ — your argument is almost correct: each connected component is contractible, but there are countably many such components ($\pi_0(\Omega S^1)=\pi_1(S^1)=\mathbb Z$).

As for $\Omega^n S^n$, these spaces have very non-trivial homotopy type: $\pi_k(\Omega^n S^n)=\pi_{n+k}S^n$ and there are a lot of non-trivial homotopy groups of spheres (say, $\pi_1(\Omega^2 S^2)=\pi_3(S^2)=\mathbb Z$, $\pi_1(\Omega^3 S^3)=\pi_4(S^3)=\mathbb Z/2\mathbb Z$ etc).

As for the Serre spectral sequence, maybe you made the standard mistake: when computing, say, $H(\Omega S^3)$ one sees that there is an element $x$ s.t. $d_2(x)=[S^3]$ and concludes that (since powers of $x$ "obviously" kill all cohomology classes from $E_2$) $H(\Omega S^3)$ "is" $\mathbb Z[x]$. But actually $d_2(x^n)=\mathbf{n}x[S^3]$, so powers of $x$ don't kill all classes: there also should exist classes $x^n/n!$ and the answer is not $\mathbb Z[X]$ but $\Gamma[x]$.