Hunt for the Giant Horse
If you travel out to the Mounted Archery Camp during the day, you can chat with the NPC Straia. The Mounted Archery Camp is west of the Highland Stable across the river.
Straia will tell you about a giant horse spotted in the Taobab Grasslands. He wants you to tame it and show it to him. Unfortunately, the horse also is evidently in close proximity with other "horse-like" monsters... This usually wouldn't be a problem, but you have to keep the horse safe if you want to get it back to Straia and/or a stable, so that puts a snag in the plan.
The horse itself will need top-grade stealth and a ton of stamina to tame, so pack up on stealth boosting food and/or armor, and have a lot of stamina to spare. Once you are ready to go, you can mark the sidequest on your map and the location of the horse will pop up.
If you go there by foot from the Mounted Archery Camp, you will run into two Lynels spread apart along the path. You should probably defeat them so you can bring the giant horse safely back to Straia because the giant horse's speed cap is lower than that of a normal horse. That means the Lynel can chase after your horse and hurt it.
The giant horse will be immediately obvious as an excessively large horse in a herd of other normal-sized horses. It also has a fiery orange mane. Take a photo of it so that it can be logged into your Hyrule Compedium, as it has a unique entry. Then sneak up on it and mash [L] until it's tame.
Once you have the horse tamed, ride it back to where Straia is waiting at the Mounted Archery Camp. The giant horse is fairly temperamental so you will likely have to soothe it a lot. On the way back you might also pass by some Bokoblins riding on horses, but they won't make chase as long as you keep moving.
After you have talked to Straia and let him look at the giant horse, you will receive a silver Rupee worth 100 Rupees for your trouble. You can then either let the giant horse go or take it to the nearby Highland Stable to be registered!
The giant horse has a wild temperament, similar to the white stallion. It also has maxed out strength, making it one of the strongest horses in the game. In exchange, the speed is only one star, so it is also one of the slowest horses in the game. Like other unique horses in this game, the giant horse cannot be found again if it dies.