The short answer is that there is no such word. The puzzle is a hoax, a misunderstanding, or a shaggy dog.
If you want an idea of how low people will stoop, one really inane explanation of this puzzle is that it should be interpreted as "There are three words in the English language..." in which case, the third word is obviously "language". Unfortunately, the other two answers are not "angry" and "hungry", so how this idiocy can satisfy anyone I'll never know.
The puzzle seems like such a simple question. You're sure you've heard such a word, and the fact that the puzzle states that there is one means that you can almost visualize it. But this is in fact a cruel hoax, that illustrates a common failing of human thought. Just like saying "Don't think of a white elephant" triggers thoughts of a white elephant, saying "the third word that ends in GRY" causes the mind to build a mental image that can't quite come into focus, but which the mind comes to believe is the answer to the question. After days of trolling through dictionaries, the bleary-eyed searcher refuses the advice of friends to give up. He may he hungry, he may be angry, but damn it, he's definitely going to be that third thing too!
Alas, the puzzle is a fake. The answer is, THERE IS NO THIRD WORD!. That's not quite true; there are a few strings of characters, which appear in some edition or other of some unabridged dictionary, and hence are "words". It's just that they're nowhere near the kind of common words that "angry" and "hungry" are.
But everyone that I tell this to is unhappy. They really want there to be a third common word ending in "gry".
So OK, here's a list of a few goofy "GRY" words. See if this will satisfy you. There are many more, but trust me, these are the pick of the crop. English has quite a boneyard we don't usually visit!
So feel angry, feel hungry, but just don't blame me! (I already spent a week paging the dictionary, and years wondering if I had missed an obvious word somehow...)