A few years ago, I bought several sets of 600 count, Egyptian cotton sheets. They wrinkled horribly. They looked like crinkled crepe paper with both small wrinkles from washing & deep creases after sleeping on them. I've seen origami paper has has creases less deep.

(Probably the reason they were on clearance.

) I'm not one to iron sheets either.
The Egyptian cotton, which I've always seen described as being incredibly soft, was TOO soft and lacked structure, instead aiding in wrinkling. As for the 600 count, I've since read in a lot of sources that higher thread count doesn't necessarily make for better, thicker sheets. There is only so many threads that can be woven into a one inch square space. If one tries to squeeze a lot more, something has to give. Often it's the threads are just made thinner to fit them all in. Sure enough, when I read reviews on higher thread count sheets, people often complain how easily the sheets shred or tear really easily.
So I went back to 300 count sheets, although I didn't want the crispy percale I had grown up with - that never wrinkled. I bought Target's Threshold brand, Ultra Soft Fitted Sheets. They are 100% cotton so they won't pill like synthetic. They are reasonably thick and they have an elastic that goes all the way around the bottom hem. So even if one doesn't have a deep mattress, it wraps around the mattress snuggly. They are soft. And best of all they don't wrinkle.

I'm really happy with them.
https://www.target.com/p/300-thread-count-ultra-soft-fitted-sheet-threshold/-/A-13973172