An example of a space $X$ which doesn't embed in $\mathbb{R}^n$ for any $n$? Apologies if this has been asked before, but couldn't find it.
The definition of embedding that I'm using is this:

Suppose $X$ and $Y$ are topological spaces. We call a function
  $f:X\rightarrow Y$ an embedding if $f$ is a homeomorphism from $X$
  to $f(X)$, equipped with the subspace topology.

I think the idea is to look for a space $X$ where any function $f:X\rightarrow\mathbb{R}^n$ does not have a continuous inverse. I can't seem to get anywhere.

To add to the answers above, to get a Hausdorff space of continuum cardinality you can take a disjoint union of countably many simplexes of unbound degree (i.e. a $1$-simplex, a $2$ simplex etc.). Since an $n$ simplex can only be embedded in $\mathbb{R}^m$ for $n\le m$ you get that their union fits the goal.
