MTG

As the name suggests, Pauper is the go-to budget format. Admittedly, it's less cheap than it used to be. But at its cheapest, it is one of the least expensive.

EDH (Commander) has a huge range in price, but some decks (check out The Commander's Quarters on YT) are super fun and affordable. It's a good casual format to start a budget deck with and continue growing. It's also a gateway drug, so be warned 😬

Lastly, if you are playing in a private pod (i.e., not at your LGS), you can set your own limits based on market price. For example, no decks over 50 bucks total, no single high-cost card over 10 bucks, etc. Ultimately, rule zero applies to budget-conscious private games. You decide what's allowed.

I don't know if it's the cheapest but I recently got back into magic after 15 years and my friends have been playing eternal brawl.

It plays really well with 3-4 players but works for 2-6. Like commander you have a commander but only 60 cards and all of your cards need to be from a block to avoid bad combos.

It's possible to set up a deck for a few dollars if everyone is avoiding expensive cards and using a few sets old block usually means most cards are a few cents.

https://eternalbrawl.wordpress.com/formats/