Non-algebraic Lie groups

Question

When trying to learn about Lie groups I find that most natural examples of Lie groups are actually examples of algebraic groups.

What are some interesting examples of Lie groups which are not algebraic groups?

Answer 1

This post was incorrect. However, as Theo has pointed out, the double cover of $SL_2(\mathbb{R})$ is an example of a Lie group which is non linear and non-abelian, and hence, if it is algebraic, it is at least neither affine nor projective.

The wikipedia page on Linear algebraic groups has a list of a few criterea which prevent a lie group from being an algebraic group.