For the unfamiliar, Abserd is a well-known story of a character who took one level in every class in D&D 5e. Inspired by a friend of mine who played a form of this idea in a one-shot some years ago, I've put together a build that maximizes your number of pets while never taking more than one level in a single class.

The Build: Druid (Pack Lord) 1 / Cavalier 1 / Hunter (Packmaster) 1 / Samurai 1 / Alchemist (Winged Marauder) 1 / Barbarian (Mad Dog) 1 / Ranger (Falconer) 1 / Vigilante (Mounted Fury) 1 / Warpriest (Divine Commander) 1 / Oracle* 1 / Sorcerer (Wildblooded)** 1 / Brawler (Wild Child) 1 / Inquisitor (Sacred Huntmaster) 1 / Shaman (Draconic Shaman) 1 / Cleric (Undead Lord) 1 / Asavir 1 / Sacred Sentinel 1 / Summoner 1 / Spiritualist 1 / Wizard*** 1

*Nature mystery, Bonded Mount revelation

**Sylvan bloodline

***Arcane bond: Familiar

Pack Lord Druid and Packmaster Hunter mean that you can divide your effective druid levels amongst any number of animal companions, up to your effective druid level.

The Druid, Cavalier, Hunter, Samurai, Alchemist, Barbarian, Ranger, Vigilante, Warpriest, Sorcerer, Brawler, Inquisitor, and Sacred Sentinel levels each grant you one effective druid level, as does the Bonded Mount revelation on the Oracle. The Asavir level actually grants you three effective druid levels, up to a maximum effective druid level of your character level. That adds up to effective druid level 17.

The Shaman level grants you a drake companion in place of the typical spirit animal. The Cleric level gives you a corpse companion. Summoner gives an eidolon and Spiritualist gives a phantom (unfortunately these can't be summoned at the same time but they still count as two pets). Wizard, obviously, gives a familiar.

All in all, that adds up to 22 pets, 21 of which can exist at one time, without ever taking more than one level in any class. You also have a base attack bonus of 7 at twentieth level, and spellcasting from 11 sources with nary a 2nd-level slot, and none of your pets are advanced beyond 1st-level capabilities (unless you took Boon Companion to bring some of them up to 5th). So uhhhhh. Watch out!