We say that a continuous function $f:\mathbb{R}\to\mathbb{C}$ is almost periodic in the sense of Bohr if:
For every sequence $(t'_n)_{n\geq0}$, there's a sub-sequence $(t_n)_{n\geq0}$ such that $f(t+t_n)$ converges uniformly in $\mathbb{R}$ to a function $g(t)$ i.e. $$\sup_{t\in \mathbb{R}}|f(t+t_n)-g(t)|\to 0, \ \ when \ \ n\to +\infty.$$
This class of functions was proved to be the closure under the uniform norm in $\mathbb{R}$ of the class of Trigonometric Polynomial with independent frequencies $$P(t)=\sum_{k=0}^N a_ke^{ib_kt}$$
Almost periodic functions are uniformly continuous and this class contains the class of periodic functions, but functions like $f(t)=\sin(t)+\sin(t\sqrt2)$ are almost periodic but not periodic. We can show this by assuming it is periodic then get the contradiction $\sqrt2 \in \mathbb{Q}$.
A more general class is the class the almost automorphic functions. A continuous function $f:\mathbb{R}\to\mathbb{C}$ is almost automorphic if
For every sequence $(t'_n)_{n\geq0}$, there's a sub-sequence $(t_n)_{n\geq0}$ and a function $g$ such that for each $t\in \mathbb{R}$ $$|f(t+t_n)-g(t)|\to 0, \ \ when \ \ n\to +\infty.$$ and $$|g(t-t_n)-f(t)|\to 0, \ \ when \ \ n\to +\infty.$$
This class is proved to contain the class of almost periodic functions. A classic example of almost automoprhic function which is not almost periodic is: $$f(t)=\sin\left(\frac{1}{2+\sin(t)+\sin(t\sqrt2)}\right)$$ To prove this, we only have to show that this function is not uniformly continuous(since almost periodic functions are uniformly continuous). But It seems like I cannot prove it. I also cannot prove that this function is almost automorphic using the definition.
You must see (read) the following work (of Bolis Basit and Hans Günzler, posted at arxiv):
http://arxiv.org/abs/1006.2169
There you can find nice steps to prove the desired claims.
Best.