Not really rare, but I’ve seen the so-called coyote/wolf hybrids moving into the east coast, moose up way too close and personal (they are BIG), beaver, mink, and a brief glimpse of a cougar.
A very angry bobcat from about 6 feet away. Stuck in a leg hold trap. The dog almost blundered right into it. The law is that they have to check their traps every couple days, so I hope he wasn’t in there too long.
Not really a rare animal but I’ve only seen an opiliones (daddy long legs to some of you, but to me that’s a spider) just once. It was bright red.
I have petted a cougar cub. It was being carried around by its handlers while they shopped at the nursery where I worked. I guess cougars are not that rare, but the close experience was. I have also been nibbled by a tiger and kissed by a skunk.
The tiger was a cub in a zoo. We were allowed to reach in and pet them, and one nipped my finger. The skunk was a de-scented pet at a Boy Scouts camp.
A Gray Whale.
Comparatively unusual, a good number of Californian semi-wild peacocks.
I was once at a secluded beach when a penguin came hopping up out of the waves, waddled up the sand and onto the bank and nestled in. I admit I can’t remember what kind of penguin it was, but based on location there’s a strong likelihood it was a yellow-eyed, the rarest penguins in the world.
I’ve lived in Colorado for 25 years and spend a lot of time in remote areas of the mountains and have seen exactly one.
Moat unusual has to be a platypus, which I got to watch for a few minutes hunting in a stream in Tasmania, simply because nothing is weirder than them 
Frankly I have no clue what the rarest creature I’ve seen in the wild is, it partly depends on how you define ‘wild’. I’ve worked with a few very rare animals In captivity.
The rarest animal I’ve seen in the wild is the bog turtle. The most unusual, and becoming almost as rare, is the hellbender.
Unca Google says the English for urogallo is capercaillie; apparently they’re also called wood grouse. We had a disoriented-looking one walk into the factory once, eventually we were able to point him back at the door without scaring him. While not terribly rare worldwide, the lower reaches of Moncayo aren’t even marked in the map as having them (it’s slightly more than 100km south of the spot on the Pyrenees).
I haven’t seen any in ages, and again they’re not rare worldwide, but according to our 9th grade Science teacher “hummingbirds only exist in the new world”. We rebutted it, as all of us had seen some at one point. There was a family which used to come by my mother’s terrace, I had a T-shirt that they mistook for actual flowers (tickled!). That they originated in America, we had no doubt: but that they were present in the Ebro Valley and reproduced, we also had no doubt.
A possible lynx once, but we couldn’t check. Guide said it was a lynx; all we can say is that it was some sort of golden-brown large cat, and didn’t stop to pose.
In Costa Rica we saw carey turtles, which are considered critically endangered.
2 years ago I snapped a picture in the Rockies of a beaver sitting on a high outcrop. On Facebook I captioned it “majestic beaver.”
Not rare, but unusual - at least for me.
Mongooses in Hawaii. We saw one up close, because it was semi-tame. At a botanical garden on the big island, a local man was mending a fishing net at the water’s edge. He had an open plastic container of poke by his side. A little mongoose hid in the plants behind him, and came out now and then to be fed a bite of poke from a pair of chopsticks held out to him by the net mender.
My wife and I go to Barbados every winter and while there we regularly see wild mongooses (the locals call them mungos) and wild green monkeys. The latter are old world monkeys brought by early ships. One day we were sitting on our porch with a bunch of bananas in the middle of our lunch table and suddenly a monkey appeared, grabbed the bananas, ran up a nearby tree and proceded to peel and eat them all. What we took to be his family were at the base of the tree chattering up to him, but he shared nothing with them and dropped the peels on their heads.
I’ve never seen any of the cats of the Americas in the wild. I lived in Alaska for a total of 32 years and never saw a lynx. I saw moose, caribou, dall sheep, wolf, fox, grizzly, polar, black bear, musk ox, but never a cat, even when the rabbit population was booming.
One of the coolest birds we saw in Africa was the lilac breasted roller.
Hmm, nothing super rare. I’ve seen minks, and bald eagles (and eagle chicks) in new Hampshire. I’ve seen a bobcat in my backyard. And once i saw a large rabbit thing that was probably an unusual hare, or maybe it was just unusual in my back yard.
In Banff, I saw lots of large game, including mountain goats and bighorn sheep.
I saw an orchard oriole in a tree.
The term rare is open to definition. I’ve seen black rhinos and cheetahs in the wild, both of which numbered less than 10,000 left at the time. I visited mountain gorillas in Rwanda too. At the time it was estimated that there were less than 250 left, but the estimate now is closer to 900. There were probably groups that weren’t known back when I was there in 1982. I’ve also seen a newly discovered pygmy marmoset in the Peruvian Amazon, but that doesn’t mean it was rare.
Possibly the rarest critter I’ve seen were specimens of the Enos Lake stickleback. I photographed a couple around 1989. They were found elusively in Enos Lake on Vancouver island. Sadly, they have now been declared extinct. As far as I know, I have the only photos of living specimens in existence.
An ocelot while on a nature walk. They are native to Texas, but rather rare because of humans encroaching on their habitats.
Crossing the Atlantic would be physically impossible for a hummingbird - they have to eat every few hours. When they migrate, they do so over land, where flowering plants can fuel their journey.
It’s possible that escaped caged hummingbirds could have formed a viable population, or maybe you saw these. You wouldn’t be the first person to conflate them!
Chefguy, if you’re ever in the Bay Area, I’d be happy to show you “my spots” for bobcat. While not as exciting as mountain lions, they’re pretty awesome to see in the wild.
Reading this thread I was having “déjà vu all over again”. Took me a while (my search skills apparently aren’t that good) but I finally found this threadfrom 2007. Quite a few real rarities in that one! Crotalus, your wild bog turtle sighting still has me jealous! Repeating it in this thread may be what triggered my dim memory. A number of others are also impressive. I offered (in part):
Colibri is kind enough to say
Coming from him, that’s quite a statement! But I’m not so sure, as he also said
Several months later I saw an Ivory Billed in flight at a distance, in the same area. Obviously I cannot know if it was the same bird or another, but back in about 1973 there was at least one living representative of the species in Florida.
But I just briefly viewed these. New Zealand rarities would be on my ‘bucket list’ and Colibri, you didn’t just see them, you worked with them! Too cool.
We saw a lot of cheetahs and lions on safari, but only one leopard. I guess by ‘rare’, I mean endangered or seldom seen. I envy your gorilla trek. We tried three times at Bwindi on the Uganda side in 1997-98. Once, there was a snafu on the part of the booking agent and we never left camp; the other times, the gorillas had moved into Congo territory and the trek would have been at least eight hours, not to mention the danger of crossing the border. The list of animals and birds we saw in Africa is way too long for this thread, and I’m sure most other folks here will never see any of them, but that would be stealth bragging. 