None of the games I recommended are free to download. Adding cost to your criteria, I can recommend the following video games: 

1. Fortnite - A popular online battle royale game. It is a third-person shooter that involves competing as a solo player or part of a team of up to four against up to 100 other players. All players enter a large map, find equipment and move throughout the map. The map shrinks on a short interval in order to bring the remaining players closer together. The last player or team standing wins. It gets regular content updates and is free to play.

2. Apex Legends - Like Fortnite, it is another popular online battle royale game. However, it is faster-paced than Fortnite and extremely competitive. Unlike Fortnite, it is a first person, class-based game with numerous different characters with different roles and playstyles. It involves solo, duo, or teams of three to compete to be the last person or team standing. It gets regular content updates and is free to play.

3. Destiny 2 - A story-driven continuously online science-fantasy first-person shooter that allows to play with other players or largely alone. It is free to play though some content is locked behind season passes. It is renown for its three distinct classes, vast array of equipment, and frenzied gameplay though it does involve some small occasional puzzles to progress. 

4. League of Legends - A popular free to play Multiplayer Online Battle Arena (MOBA) game. League of Legends is a top-down action game with a large number of distinct classes involving two teams of up to five players fighting for territory with the intent to destroy the other teams' base. It is incredibly fast-paced, nuanced, and competitive with a well-established community of players involving up to world championship matches. 

5. Rocket League - A free to play online sports game involving, essentially, playing soccer with stunt cars in an enclosed stadium. The go It is fast and extremely competitive online due to the skill of the existing playerbase, though you can play it offline as well.
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