I'm looking for an example.
Consider defining the measure of a set $E \subseteq \Bbb{R}^d$ by a limit: Take $$m(E) := \lim_{N \to \infty} \frac{1}{N^d} \cdot \left| E \cap \frac{1}{N} \Bbb{Z}^d \right| $$
$\Bbb{R}^d$ denotes the cartesian product of d copies of the reals. Similarly define $\Bbb{Z}^d$ for the integers. Also I think $N^d$ is a cartesian product of $d$ copies of the number $1/N$.
What is an example of a set $E$ where this limit does not exist?
If you consider $d = 1$ and the subset of the rationals in $[0,1]$ that have exactly one pure prime power in their denominator, then you will get that the limit does not exist if $N \to \infty$ can be any integer, i.e. as $N$ increases the value will fluctuate between $0$ and $1$.
To see this, first consider when $N$ is prime, and then consider when $N$ is a product of the first $k$ primes for increasing $k$.