I was ask to respond to this thread but you need to check as my knowledge is "dated". However; from memory we hunted Germany, Scotland, Argentina, New Zealand and Eastern Canada without a guide.
It looks like all the responders, including myself, is assuming your asking about other Countries, and not States in the U.S. There are opportunities to hunt various species in some states within the U.S. without a guide.
Also, "opportunities of a lifetime" is a pretty large and open ended question. If you could narrow that down a bit--Grizzly, 60 "" Moose, Goat, Sheep, Cape Buffalo, Doves, etc --the gentlemen here would be in a better position to "guide" you to where ( or if ) you can do that without a guide.
Speaking only for myself, there are more advantages to having a good guide than just the opportunity of getting on a good animal. For me, it was like a vacation, not to have to cook, clean, etc.
As others have said New Zealand is a good option. Might have to wait a year or two due to covid restrictions. Getting a rifle in can be a pain. But other than that is really is very easy to hunt here - no tags or seasons. I always recommend Tahr and chamois as they are plentiful and most kiwis stick to deer.
On my last trip to NZ in 2019 right after the mosque shooting, I had no problems bringing my A5 shotgun and bolt action rifle.
However, they made a big deal about my 1 box of 300 win mag ammo. They said nothing about my 12 ga ammo.
They passed me on 4 times to different authorities wondering what to do about it. Finally the 4th person let me go.