Pauper is a very cheap way to play Magic, 60 card decks with commons only. Some of the top competitive decks only run about $30, and you can build fun decks up with 10 cent cards from your LGS.

For casual play, my favourite is still Commander though. You can definitely build commander decks on a budget (one of the ones I made run me $26) and the format is a lot of fun!

It's not the absolute cheapest, but I recommend trying out commander if you have not already done so. The value of the preconstructed decks is very high as they come with a bunch of useful rares. Some of these decks are as low as $20 on Amazon. Additionally, almost every card ever made is allowed to be played, and nothing ever rotates out like in standard format. It's a casual format so if you don't have a lot to spend on perfecting a deck, many people have several tiers of decks for more fair matchups. If it's just you and your friends, most of the preconstructed commander decks are well matched. Also there are 100 unique cards in each deck, so there is a ton of variety even in one deck and it will take many games before you even see all the cards you have.

If you're just playing casually I recommend you get proxies printed. That's the absolute cheapest way to play any format you want.

You can print your own at home, or if you want something similar in quality to actual cards I suggest you check out MPC. There's also a really handy website that makes the whole process of ordering from MPC very easy.

Pauper is the obvious answer. It's unique in that every card in the format is Common rarity, so that mitigates price already. Check out MTGGoldfish's Metagame page for Pauper, which lists competitively viable decks and their price. Not sure how up to date it is, but it's a good starting point!

Pauper would be a good choice! And there is also a brand new place for it here :)

The cheapest format that is somewhat widely played is Pauper. All cards that have been printed as common once are legal, most competitive decks are sub 100€/$. Casual even cheaper! I like it a lot, the card pool is deep and the meta somewhat balanced, except for affinity maybe.

Sharing your cards by shuffling up a pile of cards and "drafting" from them is the cheapest. Since the cards are shared you get rid of the "pay to win" problem.
You don't all need to share all cards you have, but having a "draft cube" like this makes it less of a collectable and more like a board game. You can get cheap or even free "draft chaff" cards at many game clubs.

I recommend Commander for playing in a group, since it has lots of fun gameplay, there are plenty of powerful budget lists and basically all the cards ever printed (with very few exceptions) are legal, so you can play what you have and never have to worry about rotation. I recommend looking up some budget lists to start out with, as they are generally significantly more powerful than preconstructed decks (some exceptions apply, but those precons are no longer cheap), while not being all that much more expensive.

Alternatively, as others have suggested, Pauper is a fun format to play that is basically guaranteed to stay fairly cheap, due to all the legal cards being printed at common at some point.

I recommend looking through (budget) decklists for either format with your friends and then agree on which one looks more fun for your group.

My and my friends always liked brewing our own decks so to encourage that and discourage net decking we've come up with our own casual budget format we call Bargain. The only special rules are your maindeck must be $30 or less on mtg goldfish and your leftover money plus an additional $5 can go toward a sideboard. We just check every couple of months whether a deck has gone way over budget and we haven't had the need to ban or restrict any cards yet.

Thanks everyone for your comments, I really appreciate them all!

What do you expect from those games? Do you want to play magic or are you going for a free for all commander experience?