I can think of the set of bounded, continuous functions from $\mathbb R \to \mathbb R$ as a group, with composition as addition of functions. In other words, this group has the rule that the composition of two elements in the group, $f(x)$ and $g(x)$, is their point-wise sum $f(x) + g(x)$. Since this set is path connected, the group is continuous, i.e. it is a lie group.
What is the lie algebra of this group?
I am trying to use this approach to find a relationship between Lie theory and Fourier series, but I realize this might not work. So I have decided to just ask the question in this form. If anyone knows of any connections between Lie theory and Fourier series however, I am interested to hear about that.
(I realize I may have made some wrong/poorly stated assertions, please point these out in comments)
Since your Lie group is infinite dimensional, one has to be careful about what exactly you mean by Lie algebra, and so on.
In any case, your «Lie group of bounded continuous functions $\mathbb R\to\mathbb R$ under composition» is not really a group... You'd have to restrict your attention to the group of homeomorphisms (or something along that line) to actually get a group.
The last sentence of your first paragraph is a bit misguided. You could do much worse that read Warner's book, say, on differentiable manifolds and lie groups, or Lang's, if you like infinite dimensional situations.