Does LEGO Battles Support 3-Player Mode?

No. LEGO Battles (DS, 2009) officially supports only 1-player (Campaign) and 2-player (Local Wireless / Nintendo Wi-Fi Connection) versus modes.

Details

• Campaign: 6 factions, 30 missions, single-player only.
• Versus: 2 players via local wireless (DS-to-DS) or defunct Nintendo WFC. Map selection, army building.
• No 3- or 4-player support in code.

Unofficial Mods

Homebrew community (nds-homebrew, GBAtemp) has experimented with multiplayer patches, but none stable/released for 3+ players. Emulator netplay (melonDS, DeSmuME) only replicates 2-player local wireless.

3-Player LEGO Alternatives

• LEGO Star Wars: The Complete Saga (Wii/PS3/360/PC) – 2-player co-op only.
• LEGO City Undercover (Wii U/Switch/PS4/Xbox/PC) – 2-player co-op (Switch/PS4/Xbox/PC).
• LEGO Worlds (Switch/PS4/Xbox/PC) – 2-player online co-op.
• LEGO Brawls (Apple Arcade/Switch/PC) – 4-player battle royale.

No LEGO RTS supports 3 players natively.