I confused myself here. Let $W_t$ be Brownian motion. It is obvious that the variance of the Brownian motion increment $dW_t$ is $dt$, i.e. $E[(d W_t)^2] = dt$. However, if I write this like so, I get a different result:
\begin{align} E[(dW_t)^2] &= E[(W_{t+dt} - W_t)^2] \\ &= E[W_{t+dt}^2] - 2E[W_{t+dt}]E[W_t] + E[W_t^2] \\ &= (t+dt) - 0 + t \\ &= 2t + dt \end{align}
Where is my mistake in the expansion above?