There is the Barratt-Milnor 1962 example of "anomalous (singular) homology", showing that the rational singular homology of the one point union $X$ of countably many spheres $S^2$ whith radius tending to $0$ is non zero in all dimensions $>1$ (and is in fact uncountable). They use Hurewicz maps and infinite sums of Whitehead products of elements of homotopy groups of spheres, but I don't see if torsion in higher $\pi_i(S^2)$ could give torsion in $H_*(X,Z)$.

There is the Barratt-Milnor 1962 example of "anomalous (singular) homology", showing that the rational singular homology of the one point union $X$ of countably many spheres $S^2$ whith radius tending to $0$ is non zero in all dimensions $>1$ (and is in fact uncountable). They use Hurewicz maps and infinite sums of Whitehead products of elements of homotopy groups of spheres, but I don't see if torsion in higher $\pi_i(S^2)$ could give torsion in $H_*(X,Z)$.