Golf me a card dealer

I think what follows gives an "arithmetic" example of what you're looking for: a variety $X$ over a number field $k$ such that $X$ is $\mathbb{Q}$-factorial but $X_L = X \times_k L$ is not, for some finite extension $L/k$. In other words, $X$ is $\mathbb{Q}$-factorial, but not locally $\mathbb{Q}$-factorial in the étale topology.

Take an elliptic curve $E$ over $k$ which has rank zero over $k$, but positive rank over $L$. Embed $E$ as a plane cubic, and let $X$ be the projective cone over $E$. Then the only Cartier divisor classes on $X$ are multiples of a plane section. Now there is an isomorphism $\mathrm{Cl}_0(X) \to \mathrm{Cl}_0(E) = \mathrm{Pic}_0(E)$ (Hartshorne, II, Ex. 6.3). So the fact that $E(k)$ has rank zero means that $\mathrm{Cl}_0(X)$ is finite, so $\mathrm{Cl}(X)$ has rank 1, so the quotient $\mathrm{Cl}(X)/\mathrm{Pic}(X)$ is finite. Thus $X$ is $\mathbb{Q}$-Cartier. On the other hand, over $L$, $\mathrm{Cl}(X)/\mathrm{Pic}(X)$ has positive rank. Specifically, let $P$ be a point of infinite order on $E(L)$; then the line over $P$ gives a Weil divisor on $X_L$ no multiple of which is Cartier.

I imagine the same argument, starting from a elliptic surface over $\mathbb{P}^1_\mathbb{C}$, could give a "geometric" example.

Why not imagine the third-order process as a two-stage second-order process like so:

I like Brandon's very physically intuitive answer: mine is a little drier. It is simply that three waves $E_j(t);\,j=1,2,3$ mix through $n^{th}$ order nonlinearity by way of $n^{th}$ power term $\left(\sum_{j=1}^3 E_j(t) e^{-i\,\omega_j\,t} + E_j(t)^* e^{i\,\omega_j\,t}\right)^n$ in the Taylor series for the input to output transfer function. So in the $n^{th}$ order term we get frequencies

$$|\sum_j a_{n,j} \omega_j|\qquad(1)$$

where:

$$\sum_j |a_{n,j}| = n;\quad a_{n,j} \in \{0,1,2,\cdots n\}\qquad(2)$$

So you need, as you can guess, at least third order to get terms where all three frequencies are together in the sum. From (1) you'll get:

$$\begin{array}{c} |\omega_1 \pm 2\omega_2|\\ |\omega_1 \pm 2\omega_3|\\ |2 \omega_1 \pm \omega_2|\\ |2 \omega_1 \pm \omega_3|\\ |2 \omega_2 \pm \omega_3|\\ |\omega_1 \pm \omega_2\pm\omega_3| \end{array}$$

the last term is the only third order one where all three frequencies combine. You need to go to four wave mixing and higher order to get more "interesting" linear combinations of frequencies of the form being $|2 \omega_1 \pm \omega_2\pm\omega_3|$ and fifth order to get $|3 \omega_1 \pm \omega_2\pm\omega_3|$.