I'm pretty sure that most Eggs adopted or bought from various Locations in-game are left un-tested by default, as another had said in this Thread. Yet, below is the following ones I've found while scanning different locations.

Umbra Forest:
- Aryth's Spire: Tamed Hatchling Faelora
- Aryth's Spire: Tamed Egg
- Faerina's Shrine: Tamed Hatchling Qitari
- Garden of the Magnates: Tamed Egg
- The Forgotten Tomb: Tamed Hatchling Vulnyx

Viridian Meadows:
- Trollsblight Caverns: Tamed Egg
- Fungus Grove: Tamed Egg
- Nightfall Bandit Camp: Tamed Egg

Astryl's Vale:
- Puffadore Pet Shop: Tamed Hatchling Puffadore

Note: Trapped Tamed in these Areas so far seems to be auto-tested upon capture as intended.


Other Areas:
- Scale Shop: Tamed Egg

The Forsaken Guardian Ny'vene Egg from Dungeon of the Forsaken are not automatically tested... This one is because I did it, when I saw there was not automatically. ^^'

Lucky Pot egg won today.

https://www.sylestia.com/view/pets/?petid=7690732

The BTB Bulb egg from the egg pop-up.
https://www.sylestia.com/view/pets/?petid=7688061

As well as eggs from the Temple in TLG.
I did test them yesterday, as I was curious about the stats.
https://www.sylestia.com/view/pets/?petid=7688043
https://www.sylestia.com/view/pets/?petid=7688037

found this in the forgotten caverns
https://www.sylestia.com/view/pets/?petid=7690033

this themed egg found in the lost grove was untested:
https://www.sylestia.com/view/pets/?petid=7690921
as well as this wild spawn from sweet apple orchard:
https://www.sylestia.com/view/pets/?petid=7690873
and this found egg from astryl's tranquility (EDIT: this tested automatically upon hatching):
https://www.sylestia.com/view/pets/?petid=7690900
also this themed wild spawn from the lost grove:
https://www.sylestia.com/view/pets/?petid=7691238
along with this non-themed lost grove wild spawn:
https://www.sylestia.com/view/pets/?petid=7691462

(EDIT: i ended up manually testing all of these eventually, though the one was auto-tested upon manually hatching)

Eggs found in the Dungeon of the Forsaken are untested:
https://www.sylestia.com/view/pets/?petid=7690318
https://www.sylestia.com/view/pets/?petid=7690320
UPDATE: those two eggs hatched on their own, and they were still untested as hatchlings. I had to test them manually.

Egg pop-ups eggs are not tested either:
https://www.sylestia.com/view/pets/?petid=7690336
https://www.sylestia.com/view/pets/?petid=7691080
UPDATE: the first egg above (Fortuitous Wanderer) was hatched manually by me, and was tested automatically.

I have this pet here, from an egg found in the Lost Grove - it was untested, but I think it was tested upon hatching: https://www.sylestia.com/view/pets/?petid=7684810

This Ny'vene I received from the Fortune Event Login Rewards was not automatically tested and was not auto tested after hatching. I did not hatch it myself, I didn't log in for a while and it hatched automatically. I tested it myself once I realized it.

Eggs from non-Ethernia areas don't seem to be auto-tested. Can't remember exactly what zones I got these from but they weren't tested.
https://www.sylestia.com/view/pets/?petid=7688147
https://www.sylestia.com/view/pets/?petid=7688151
Pets I got from the secret maps were not tested either, but I've already tested them manually myself.

https://www.sylestia.com/view/pets/?petid=7687059
https://www.sylestia.com/view/pets/?petid=7687064
https://www.sylestia.com/view/pets/?petid=7687072
also non lg eggs, i know one of them is from aryth's spire, and one is from sweet blossom orchard