41 life without using Black Lotus

This is a legacy/modern entry (i.e. doesn't use Black Lotus).

Reveal Providence to go to 26 life. Then exile Autochthon Wurm to cast Nourishing Shoal to gain 15 life.

Multiplayer: 20 + 9 * number of opponents

The question clarifies that there are two players. If we relax that restriction, we can do better in cases where there are 4 or more players. Have 3 Chancellor of the Dross in your opening hand and reveal all three to steal 9 life per opponent.

4 Cards: Arbitrarily high life

The question specifies three cards for a reason. Allowing one more card lets you go infinite. The 4 cards are Black Lotus, Daru Spiritualist, Shuko, and Diamond Valley.

Use Black Lotus to Cast Daru Spiritualist and Shuko. Repeatedly equip Shuko to Daru Spiritualist, giving Daru Spiritualist +2 toughness each time, until has [arbitrarity high odd number] toughness. Play Diamond Valley and use it to sacrifice Daru Spiritualist to gain [arbitrarity high odd number] life.

Black Lotus x2
Beacon of immortality

Depends on the format how much life you gain. 20 on a default game for example or 40 in Commander for example

The best I can manage is 32 life:

• Black Lotus
• Phyrexian Dreadnaught

• Sheltering Word

  1. Cast Black Lotus and sacrifice for 3 green mana.
  2. Cast Phyrexian Dradnaught.
  3. In response to the etb, cast Shltering Word. [gain 12 life]

A second solution for 32 life:

• Black Lotus
• Ritual of Restoration

• Weather the Storm

  1. Cast Black Lotus and sacrifice for 3 white mana.
  2. Cast Ritual of Restoration, returning Black Lotus to hand.
  3. Cast Black Lotus and sacrifice for 3 green mana.
  4. Cast Weather the Storm, creating 3 copies with storm. [gain 3 life * 4]

Black Lotus x 2, Auriok Salvagers, Stream of Life

Play the lotuses and crack them for white. Cast Salvagers, use the 2 leftover to return lotus.

Cast lotus, crack for white, return lotus -> repeat as much as you want to gain a white mana each loop.

Return lotus, crack for green mana, cast stream of life for x + arbitrarily large number .

With just 2 cards, you can be at arbitrarily many life points.

The 2-card combo is Black Lotus and Game of Chaos. As long as both you and your opponent keep playing, the stakes increase indefinitely, and you can end the game whenever you win a flip.

Assuming both you and your opponent keep playing exactly until someone gets to zero life or below during Game of Chaos' resolution, you can go to 103 life with the 2 card combo, or 109 with 3 cards.

If you win (+1), lose (-2), lose (-4), win (+8), lose (-16), win (+32), win (+64), then you (and your opponent) will be at

• 21 (19),
• 19 (21),
• 15 (25),
• 23 (17),
• 7 (33),
• 39 (1),
• 103 (-63) life.

At the start of the game, reveal Providence for another 6 life - this doesn't change the sequence of coinflips because your opponent would still fall to 1 life at the same point of the sequence.

I'm going to cheat a bit... because you didn't specifically state that this was a 2-player game only (though you did use the word "both"...)

Hand:

• Black Lotus
• Black Lotus
• Blood Tithe

This gives you an unbounded 3n life, where n is the number of opponents you have in the game.

With only 2 cards, you can get 2n life using Syphon Soul instead.

Here is my best solution (42 Life Points):

3 Cards:

• Black Lotus
• Illusions of Grandeur
• Radiant Fountain

Turn

1. Pass until Precombat Main Phase
2. Play Radiant Fountain [gain 2 life]
3. Cast Black Lotus
4. Activate Radiant Fountain
5. Activate Black Lotus's ability, adding
6. Cast Illusions of Grandeur [gain 20 life]
7. Pass until End Phase