I'm trying to come up with $x'=Ax$, which is a system of linear differential equations, whose flow satisfies $\lim\limits_{t\to\infty} \lvert e^{tA}x\lvert = 0$ for all $x\in \mathbb{R}^n$, but $\lvert e^{tA} x\lvert$ is not monotonically decreasing.
I thought that in order to come up with an example of such a system, one would have to find a matrix $A$ with complex eigenvalues with negative real parts. But now I'm not so sure that this kind of an example would be correct since $e^{\Re e\{\lambda\}t}$ decreases much faster than the $\cos$ and $\sin$ terms would oscillate upwards. Yet, if we look at an example of the graph of such an oscillation, it usually does not decrease monotonically.
So where's the "catch"? I guess I'm conceptually wrong somewhere in my understanding of how $e^{\Re e\{\lambda\}t}$ dominates the trigonometric terms.
It is not necessary to consider nonreal eigenvalues. It is in fact sufficient to consider nontrivial Jordan canonical forms, such as $$ A=\begin{pmatrix} -1 &1\\ 0&-1 \end{pmatrix} $$ The equation $x'=Ax$ has the properties that you want (draw the phase portrait or compute explicitly the solutions).
On the other hand, it is always possible to introduce a norm such that the map $t\mapsto \|e^{tA}x\|$ is strictly decreasing for all $x\ne0$.