Here are three nontrivial examples with uniformly elliptic equations.
1 Nadirashvilli and Valduts example of a solution to a fully nonlinear uniformly elliptic PDE with constant coefficients which is not $C^2$.
Nadirashvili, N. and Vlăduţ, S.
Nonclassical solutions of fully nonlinear elliptic equations.
Geom. Funct. Anal. 17 (2007), no. 4, 1283–1296.
The original example was in dimension 12. In a later work (joint with Vladimir Tkachev) they brought the dimension down to 5. The example is known to be impossible in dimension 2. Dimensions 3 and 4 are still open.
2 Safonov's example of a uniformly elliptic equation in 3D whose solution cannot be Lipschitz continuous.
Safonov, M. V.
Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients. (Russian) Mat. Sb. (N.S.) 132(174) (1987), no. 2, 275--288; translation in Math. USSR-Sb. 60 (1988), no. 1, 269–281
Such example is known to be impossible in dimension 2.
3. Plis' example of a uniformly elliptic equation with Holder coefficients for which the unique continuation property does not hold.
On non-uniqueness in Cauchy problem for an elliptic second order differential equation.
Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11 1963 95–100.
In particular, he proves that there exists a non zero solution to some uniformly elliptic PDE in 3D with Holder coefficients which is identically zero outside of a ball.
It is related to the example that Hendrik Vogt suggested. Again, in dimension 2 it would not be possible.