Since there's a "number theory" tag, I suggest the quasimodular form $E_2(\tau)$, defined for $\tau$ in the upper half-plane as a multiple of $\sum_{m\in\bf Z} {\sum_{n\in\bf Z}}' (m\tau+n)^{-2}$ where the $\prime$ indicates omission of the term $(m,n)=(0,0)$. For even $k>2$, the corresponding sum $\sum_{m\in\bf Z} {\sum_{n\in\bf Z}}' (m\tau+n)^{-k}$ converges absolutely and yields modularity of $E_k$. But for $k=2$, switching the sums yields $\tau^{-2} E_2(1/\tau)$, which is not the same thing as $E_2(\tau)$! (But you can still recover the formula for the difference by carefully keeping track of how switching $\sum_m$ and $\sum_n$ changes the sum).