What are some examples of infinite dimensional vector spaces?

1. $\Bbb R[x]$, the polynomials in one variable.
2. All the continuous functions from $\Bbb R$ to itself.
3. All the differentiable functions from $\Bbb R$ to itself. Generally we can talk about other families of functions which are closed under addition and scalar multiplication.
4. All the infinite sequences over $\Bbb R$.

And many many others.

1. The space of continuous functions of compact support on a locally compact space, say $\mathbb{R}$.
2. The space of compactly supported smooth functions on $\mathbb{R}^{n}$.
3. The space of square summable complex sequences, commonly known as $l_{2}$. This is the prototype of all separable Hilbert spaces.
4. The space of all bounded sequences.
5. The set of all linear operators on an infinite dimensional vector space.
6. The space $L^{p}(X)$ where $(X, \mu)$ is a measure space.
7. The set of all Schwartz functions.

These spaces have considerable more structure than just a vector space, in particular they can all be given some norm (in third case an inner product too). They all fall under the umbrella of function spaces.

The two examples I like are these:

1) $\mathbb{R}[x]$, the set of polynomials in $x$ with real coefficients. This is infinite dimensional because $\{x^n:n\in\mathbb{N}\}$ is an independent set, and in fact a basis.

2) $\mathcal{C}(\mathbb{R})$, the set of continuous real-valued functions on $\mathbb{R}$. Here there is no obvious basis at all. This also has lots of interesting subspaces, some of which Hagen has mentioned.