TITLE: Examples of sequences whose asymptotics can't be described by elementary functions
QUESTION [25 upvotes]: It is somewhat miraculous to me that even very complicated sequences $a_n$ which arise in various areas of mathematics have the property that there exists an elementary function $f(n)$ such that $a_n = \Theta(f(n))$ or, even better, $a_n \sim f(n)$.  Examples include 

Stirling's approximation $n! \sim \sqrt{2\pi n} \left( \frac{n}{e} \right)^n$ (and its various implications),
The asymptotics of the partition function $p_n \sim \frac{1}{4n \sqrt{3}} \exp \left( \pi \sqrt{ \frac{2n}{3} } \right)$,
The prime number theorem $\pi(n) \sim \frac{n}{\log n}$,
The asymptotics of the off-diagonal Ramsey numbers $R(3, n) = \Theta \left( \frac{n^2}{\log n} \right)$.

What are examples of sequences $a_n$ which occur "in nature" and which provably don't have this property (either the weak or strong version)?  Simpler examples preferred.
(I guess I should mention that I am not interested in sequences which don't have this property for computability reasons, e.g. the busy beaver function.  I am more interested in, for example, natural examples of sequences with "half-exponential" asymptotic growth.)

REPLY [3 votes]: Suppose F is a field of finite characteristic and u,v,w lie in A=F[x,y]. Let a_n be the dimension of the vector space A/(u^n,v^n,w^n). Suppose a_1 is >0 and finite. Then as n grows,
(a_n)/(n^2), though bounded above and bounded away from 0 generally has an oscillating "fractal-like" behavior.