This question is an excellent example of a setting where symmetry considerations yield the answer with almost no effort. Symmetry is of common use in statistical mechanics and, in the more general setting of mathematics as a whole, one could go back (at least) to Hermann Weyl for thoughts about its role. As regards probability, some recent discussion is in Symmetry and Probability by Jill North, with some Comments on the preceding, by Branden Fitelson.
The key remark in the question at hand is that the distribution of the random vector $(U_1,U_2,U_3)$ is invariant by the permutations of its coordinates, for example $(U_1,U_2,U_3)$ and $(U_3,U_1,U_2)$ share the same statistical properties, as well as $(U_{s(1)},U_{s(2)},U_{s(3)})$ for any permutation $s$ in the symmetric group $\mathfrak{S}_3$.
Now, one is interested in the probability of the event $[\max(U_1,U_2) > U_3]$. The complementary event is $ A_3=[\max(U_1,U_2) < U_3] $ and $A_3$ simply means that $U_3$ is the largest of the three values $U_1$, $U_2$, $U_3$ (there is almost surely no tie here because the common distribution has no atom). By symmetry, the events $A_1$, $A_2$ and $A_3$ have the same probability where $ A_1=[\max(U_2,U_3) < U_1]\quad\mbox{and}\quad A_2=[\max(U_1,U_3) < U_2]. $ Since these events are disjoint and their union is the universe, their common probability is $1/3$ and $ P(\max(U_1,U_2) > U_3)=1-P(A_3)=2/3. $ One sees that everything above works for an i.i.d. sample of size $n$ based on any atomless distribution on the real line (just replace $P(A_i)=1/3$ by $P(A_i)=1/n$), and even in the more general setting of exchangeable random variables, as soon as the ties have probability zero.