Yesterday I found a mobile on the bus. The person who lost it had no clue about the person possessing her mobile. I called her dad and told him I have the mobile of his daughter. So I injected the dependency from me into him. Typically a case of the hollywood principle "Don't call us (because you can't!), we call you". Later he came and picked up his daughters phone.
I'd call that a real world problem which I solved by dependency injection, isn't it?
In my opinion, DI is not THE way to solve problems, for which we would not have another solution. Factories can be another way to solve such problems.
So there is no real answer for your question, because DI is just one way besides others. It is just a pretty hip, although very elegant way.
I really enjoyed DI when I had this DAOs, which needed an SQLMapper. I just had to inject the different mappers into the fatherclass once and the rest was done by configuration. Saved me a lot of time and LOCs, but I still can't name this a problem for which there is no other solution.