All of Pokemon Mystery Dungeon Explorer of Time/Darkness/Sky's ending themes. Yes, all 7 of them. This doesn't include Sky's bonus mission ending themes or the myriad of other sad themes in that game, though.
[youtube]https://m.youtube.com/watch?v=HSFSLPU9yO4[/youtube]
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[youtube]https://m.youtube.com/watch?v=xq03I7dXCNA[/youtube]
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[youtube]https://m.youtube.com/watch?v=9RPUjQQt1o4[/youtube]
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[youtube]https://m.youtube.com/watch?v=6b1Xp_oEakY[/youtube]
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[youtube]https://m.youtube.com/watch?v=0cE0FwvKvZA[/youtube]
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[youtube]https://m.youtube.com/watch?v=1hU4kiawrzU[/youtube]
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[youtube]https://m.youtube.com/watch?v=4t-pe9Y3_nQ[/youtube]
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Pokemon Mystery Dungeon: Gates to Infinity is also a close second. Red/Blue Rescue Team is up there, too. And I'll eat my hat if Super Mystery Dungeon doesn't have the feels, either.