In a noun adjunct phrase like word ending games, the last noun is always the "real" one - that is, it's "games" of the kind "word ending". No one would ever interpret it to mean words ending in games, or words that end games. (Those would be "words ending in games" or "game ending words".)
For example:
apple tree - a tree, of the type "apple"
car door - a door, which belongs to a car
horse race - a race, of horses
race horse - a horse, for the purpose of racing
In phrases like "word ending games", it does get a little confusing because we have three nouns, not just two, so it's not always clear what modifies what. "Word ending games" could mean "games that end words" or "games about word endings".