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TL;DR: The easiest way is to use Queen Sigma or Megalo Menya Moonspell, who both give huge Banish-Boosts. For more ways to increase the number of weapons you can banish in a single run, read below.

Character bonus
Some characters have a passive bonus to the Banish stat. These are:
- Queen Sigma (+108 Banish)
- Megalo Menya Moonspell (Legacy of the Moonspell DLC) (+108 Banish)
- Menya Moonspell (+8 Banish)
- Divano Thelma (+6 Banish)

Banish Power Up
Leveling up the Banish stat directly gives +2 Banish per Rank. Up to +10 Banish can be gained this way.

Arcana
The Arcana Silent Old Sanctuary (XX) provides +3 Banish.

Golden Eggs
Golden Eggs have a chance to permanently increase Banish by 0.2 for the character that collects it.

Seal I
Sealing weapons in the Collection menu counts towards the number of banished weapons in the run. Up to 10 weapons can be sealed with Seal I.