= Solution
Suppose the claim $H=(S_1)^2$, payable at time $1$, were replicable. Holdings chosen at time zero are constants, so the time-$1$ value of any <self-financing portfolio> has the affine form
$$
aS_1^0+bS_1=a(1+r)+bS_1.
$$
But $S_1=S_0Z_1$ has a continuous positive density on $(0,\infty)$. No affine function of $S_1$ can equal the quadratic function $S_1^2$ almost surely. Therefore $H$ is not attainable and the market is not <complete market>[complete].
The same argument applies if $H$ is viewed as a time-$T$ claim: once $H$ is known at time $1$, its required discounted value at time $1$ is fixed, and the first-period portfolio value would still have to reproduce a nonaffine function of $S_1$.
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