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.

+1 on Pauper! Just created a new community here for it as well:

https://mtgzone.com/c/pauper

I think the absolute cheapest would have to be pauper, and depending on what your flg nearby runs you may be able to find games to play with others without too much work.

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/