If I choose Irelia with a spell during a showdown, does she get plus 1 might, and does Irelia's ability stack and keep increasing her might the more I choose her each turn?

Yes, Irelia, Fervent gains +1 Might when you choose her with a spell, and her ability does stack, allowing her to gain +1 Might each time she is chosen or readied throughout the turn.

Ruling Details

• Triggering the Bonus: Irelia, Fervent’s ability states: "When you choose or ready me, give me +1 Might this turn." Choosing her as a target for a spell fulfills this condition.
• Stacking: There is no "once-per-turn" limit on this ability. Every time you choose her with a spell or ability, or ready her, the ability triggers again, and the +1 Might bonuses stack.

Sequence during a Showdown

When you play a spell that chooses Irelia during a Showdown (or any time a chain is active), the following sequence occurs:

1. Play the Spell: You play a spell and choose Irelia as the target.
2. Trigger: Irelia’s triggered ability ("When you choose... me") is added to the chain on top of your spell.
3. Resolution: Because the chain resolves in LIFO (Last-In, First-Out) order, Irelia’s ability resolves first. She gains +1 Might.
4. Spell Resolution: Your spell then resolves, applying its effects to Irelia.

Important Nuance:

• Deflect: Remember that Irelia, Fervent has the [Deflect] keyword. This means that if an opponent wants to choose her with a spell or ability, they must pay an additional [A] (Rune of any domain) to do so. This cost does not apply to you when you choose your own unit.
• Targeting: The bonus is granted because you "chose" her. If a spell or effect affects her without "choosing" or "targeting" her (e.g., an effect that says "all units on this battlefield"), her ability will not trigger.

Reference: Based on the rules for triggered abilities and the specific FAQ for Irelia, Fervent.