Credit to Harvey8irdmand for steps on how to have unlimited days. Quoted at the bottom of the guide.
WARNING: you must have all pets on you at the same time in 1 run. You can not get rid of or lose any of the listed pets below, or you will have to start over.
Just a heads up, you are not going to like this achievement. The following is a list of pets you have to get all in 1 run. I recommend Dwarf/Bard. Dwarf or elf is necessary for the achievement. I recommend dwarf because you get more health. Bard is not necessary, but it allows you easy entry into elf village and you start with music.
These are listed from most difficult to least. I recommend trying for Turtles first obtaining others as they come. Once you have a picnic basket and are a little bit strong, you can try for piper bird. I spent 300ish days trying for Turtles and Piper bird. If I didn't get both, I restarted. I restarted over 10 times before I had a magical run.
Turtles: River on the road. Fish them up. Random. Will take 100s of tries.
Piper Bird: Frozen waste in an ice hut. Choose left option. Random. Not as bad as Turtles.
Bear: Get in the forest. Elf, dwarf, or picnic basket is needed.
Baby Kraken: Find in wrecked ship while searching with Siren in a boat on the sea. Get siren with music while on a boat.
Sheep: Get quest in Elf village. Appears randomly in towns after that. Can capture with magic or shackles bought from Shady Figure in Alluris.
Rats: Appears randomly in tavern quests or on the road/cave. Need music to get.
Crawler: On the Road. Obtain during Farmers Hiding roadside event. Need Dwarf/Elf to get.
Horse: Can buy in Alluris or Elf Village. Can also tame at Oasis in dunes.
UNLIMITED DAYS (Harvey8irdman): In order to subvert the issue of the time limit, I prioritized powering myself up while also not losing any animal opportunities. I got the map to baron dread in the undercity, went to the pyramid in the dunes and got a fire dragon, immense power and freed the genie. I exposed myself to raw mana and traded a soulstone to gain magic powers from the wizard in the mountains. Then I retreived the ghostblade from the ice caves in the frozen wastes and fought Baron Dread and used the fire dragon to melt his throne. Then simply DO NOT SELECT RETIRE. Poof, no more time limit and I spent 300 hundred days getting the damn turtles.
Also DONT marry the Rat Queen it permanently loses the pet rats.
If you have recommendations or corrections, please let me know in the comments and I will add or adjust the guide. I made this for the sole purpose of just providing a list to those who needed it. As most of the pets are random, it is pretty hard to put forth a step by step guide like I did with the companion achievement.