Brilliant Diamond and Shining Pearl: Mew can be received as a gift in Floaroma Town if the player has save data from LGPE on their console.
Let's Go Pikachu and Let's Go Eevee; Sword and Shield: The Poke Ball Plus, a peripheral that can be purchased separately, can be redeemed in either set of games for a Mew. Note that each Poke Ball Plus only contains one Mew.
HeartGold and SoulSilver; Japanese Diamond and Pearl: The DNS Exploit allows you to access old Mystery Gifts, including Mew in the specified games. Please see this answer for a walkthrough and this post for a list of obtainable Pokemon.
My Pokemon Ranch: Hayley will trade you Mew for any Egg after you deposit 999 Pokemon. My Pokemon Ranch can no longer be obtained legitimately, as the eShop has closed.
Japanese Emerald: Mew was briefly available in Japanese cartridges with the Old Sea Map, an event-exclusive item that takes the player to Faraway Island. This item is no longer legitimately accessible.
Mew 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 Mew events that have since expired. These can be found here.
