Here’s a little deus ex machina for you: Gandalf’s resurrection, an event which seems to be the most blatant (and literal) example of all. As others have observed, this sort of plot development only works under a proper set of narrative circumstances. What are those circumstances? I have a few ideas.
For starters, we, the audience, usually need some sort of foreshadowing or setup in order to make the DEM satisfying. We hate to get sucker-punched with a Ridiculous Development by a storyteller. On the other hand, the need for foreshadowing seems to be proportional to the difficulty of the problem that the DEM solves and where in the story the incident occurs. As ‘Pan-Man rightly points out, Bombadillo arrives to help the Hobbits with no foreshadowing or setup at all, but the problem he solves is relatively minor and occurs at the very start of the journey.
Also, I think the story needs, whether explicitly or implicitly, to deal with spiritual or supernatural themes. If towards the end of Die Hard, a story with no spiritual component or themes, Bruce Willis were to die and be resurrected, audiences would riot. Okay, maybe not riot, but you see my point.
Here’s an interesting example from the movies: Raiders of the Lost Ark. Now the end of that movie is pure DEM. The Nazis have captured Indy and Marion, they have the Ark, and they’re prepared to open it and begin learning its secrets – all seems hopeless for our heroes. But then the “power of God” literally intervenes to melt all the Nazis and save the day. Two things are interesting about this ending. One is that it is effective – not only does it work, but you can scarcely imagine any other way for the movie to end. The other is that the reason it works (again, IMO) is because it is in line with the theme of the movie, and the filmmakers set up and foreshadow the mystical power of the Ark throughout the story.
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