Canonical examples of inner product spaces that are not Hilbert spaces?

Any non-closed subspace of a Hilbert space will do. For example, the linear span of a Hilbert basis of an infinite-dimensional Hilbert space.

For an explicit example, take $H = \ell^2(\mathbb{R})$, the vector space of square summable sequences of real numbers with pointwise operations and inner product $$\Bigl\langle \{a_n\},\{b_n\}\Bigr\rangle = \sum_{k=1}^{\infty} a_kb_k.$$ Let $\mathbf{e}_i$ be the sequence that has a $1$ in the $i$th coordinate and $0$s elsewhere. Then $\{\mathbf{e}_i\mid i\in\mathbb{N}\}$ is a Hilbert basis for $H$, and its linear span (the vector space of all almost null sequences) is an inner product space that is not complete.

Lots of natural subspaces of Hilbert spaces aren't Hilbert spaces (which is the same thing as saying they're not closed). Two examples that come to mind are the subspace of sequences in $\ell^2(\mathbb{Z})$ at most finitely many terms of which are nonzero (the compactly supported sequences) and the continuous functions $C([0, 1])$ under the $L^2$-norm. (Note that you don't have to quotient out by anything here because it's already true that a continuous function with zero $L^2$-norm is identically zero.)