Crystal: Japanese and Virtual Console copies let the player obtain the GS Ball, which may be used to trigger a battle with Celebi after beating the game. Do note that VC copies are no longer obtainable following the closure of the 3DS eShop.
Colosseum: The Japanese Bonus Disc allows the player to transfer Celebi to a Japanese GBA game after all 48 Shadow Pokemon are purified. The US version came with Jirachi instead.
Diamond/Pearl/Platinum/HeartGold/SoulSilver: The DNS Exploit allows you to access old Mystery Gifts, including Celebi in all Generation IV games. Please see this answer for a walkthrough and this post for a list of obtainable Pokemon.
Celebi is also obtainable in some side series games, but most people that ask about Mythical obtainability aren't interested in these. If this doesn't apply to you, you may find a brief overview here. Additionally, there are several Celebi events that have since expired. These can be found here.
