I am studying J. Strom's Modern Classical Homotopy Theory. In chapter 4 he proposes the following exercise
Let $ X $ be a path-connected non-contractible space.
(1) It is possible that $ X \simeq \Omega X $ ?
(2) It is possible that $ X \simeq \Omega^ 2X $ ?
My answers would be
(1) No, because otherwise $ \pi_n X \simeq \pi_{ n - 1 } \Omega X \simeq \pi_{ n - 1 } X \simeq \cdots \simeq \pi_0 X \simeq *$ and by Whitehead $X \simeq *$
(2) Yes, for example $X = \prod_{n=0}^\infty K (\mathbb{Z}, 2n+1)$
The problem is that at that chapter no Whitehead theorem nor Eilenberg-MacLane spaces are available, so there would be a more down-to-earth answer!
What is available up to chapter 4?
Chapter 1: categories and functors.
Chapter 2: limits and colimits.
Chapter 3: convenient categories of spaces, i.e. with CW complexes, their limits and something more; smash product, smash-map adjuction, suspension, loopspace.
Chapter 4: homotopies, contractible spaces, nullhomotopies, abstract homotopies with cylinder and path objects, (homotopy) groups and cogroups, homotopy groups; mapping spaces; maps over and under a space; CW structure on loopspace.