Are there any examples for $u(x_1, x_2) = \max\{x_1, x_2\}$ in real word? I know how its graph looks like, and it's like when you want to choose between 2 inferior goods you choose the cheaper one so you can have more, but is there another examples?
One way to think about this utility function is "I can only use one of these two goods, and I will use the one that I have more of"

Some examples:


Choosing between two sets of equally-valued but non-matching dishware. Once you have one set, the other one is useless to you.
If your budget constraint is such that you can only have one kind of platform of some kind, then "items to be used with that platform" works. As commented above, Android apps and iPhone apps, or multi-platform video games, or DVDs vs Blu-Rays.
If you were an opportunistic politician deciding whether to pander to group A or group B, knowing that whichever group you pick the other will hate you, then "number of voters in group A" and "number of voters in group B" would go into your utility function as max(A, B).