We’ve seen humpback whales while whale-watching off of Maui. And once while snorkelling in Hanauma Bay on Oahu, a BIG green sea turtle came to check me out.
If aramdillos are almost extinct, that would have to be because of them all getting run over by cars in Texas. 
Nothing very rare here - I’ve seen tons of bald eagles, in Alaska I saw a grizzly bear with three cubs, I’ve seen sea turtles and golden eagles and red shouldered hawks. Once I saw a fox, not an exciting fox, just the regular kind. Nothing really rare.
forgot to mention hawksbill sea turtle (endangered). We had one come in during a field experiment and eat all our samples. 
Got great video of it, though.
We saw bobcats hunting several times in the Pt. Reyes area north of San Francisco. There’s also a herd of Tule Elk around there, but they aren’t hard to find.
I’ve seen a fox, a bald eagle, and a beaver, off the top of my head, but the biggest one I can remember:
A bunch of friends and I on a school trip are skiing along the trails out into the woods, following in each other’s tracks. We’re kicking up snow, sliding along, falling over and having a good time, when we round the corner.
Standing right there, in the middle of the snowy white forest, is a huge deer looking back calmly at us, without a flicker of an eyelash. We stare in complete silence, with the snow falling all around us, until it lopes off.
Talk about magic. 
I once saw three blue whales while flying at about 1000’ over the ocean off the west coast of Australia. I have seen many humpback whales but they definitely aren’t rare here, we are on the annual mating/migration route.
I’ve also seen whale sharks while flying down the west coast, though they are distinctly unimpressive when compared to a blue whale.
While living in New Zealand I saw yellow eyed penguins regularly and the occassional royal albatross.
I’ve seen beaver, moose, loons and so many bald eagles I don’t even look up anymore (I do look when they dive bomb for a fish ten feet away from me, I may also wet myself a bit).
The coolest thing for me was an otter that followed us through two lakes in the boundry waters. It would swim under our canoe, pop up and look at us, then run up on the shore over and over again until it apparently found something more interesting and took off.
Hawaiian Monk Seal, beached.
Probably later died. 
A reasonable number of endangered Hawaiian birds.
Well, given that many of the species to be found on the Galapagos Islands are unique to not only the isles as a group, but often to that particular island itself, I would have to say a lot of the birds & animals I saw there qualify.
Why do you think so? Seals go up on the beach all the time. Was it just in an odd area?
That reminds me that we also used to look down on beaches full of Stellar’s sea lions at Pt. Reyes. Big fat honkin’ loud guys.
I vacationed in the Florida Keys one summer and we saw lots of Key Deer while we were there. We had a couple even come into our camp early in the mornings and late in the evenings. According to the Wiki article there are between 300-800 left.
I saw an albatross once, flying over the water while I was standing on the beach in Virginia. There aren’t supposed to be albatross(es?) in the North Atlantic, so I can only surmise it must have gotten blown off course by a storm or something. I have heard reports of Yellow-Nosed Albatross being seen off the coast of NC, and the photos I’ve seen of them look an awful lot like the bird I saw, so maybe that was it.
I saw a mongoose in Hawaii.
I’ve seen Southern Right whales and Giant river otters in the wild.
a bald eagle perched in a tree on the way to my IL’s. We live in a grassy valley at the foot of mountains, so our area is full of raptors. Lots of osprey, hawks, eagles, etc.
I’ve also seen a brown bear in the wild. Fortunately, it was over a mile away below the trail we were hiking on.
I’m thinking the North American Bison I saw at Yellowstone doesn’t count. 
You’re shittin’ me. That’s the rarest of the rare. A bird thought to be extinct! Too bad you didn’t get a photo. You would’ve been famous. If you know in your heart that you saw it, that’s good enough! Gotta get that book, it sounds great.
That’s awesome seeing a condor BEFORE the captive breeding program. I wonder how many there were left at that time.
Heh, I’ve seen Blue Whale Poop. It’s pink because of all the krill they eat. Blue Whales started showing up in increasingly large numbers in the Santa Barbara Channel around 1991. Before then, it was one of the rarest whale sightings. Now, if you go during the peak season, you are pretty much guaranteed a sighting. I saw about 8 of them the last time I went out. I tell everyone to go on the whale watching tour because it is one of the most awesome things you can see. That is one big friggen’ fish.
Extremely rate, but I’ve seen a couple. It helps that I was a TA at UCSB. There’s usually at least one per graduating class, even back in the 80’s. They all go into marketing.
I hope they’re not too rare. I’ve got a ukelele from Ecuador that has a body made from an armadillo.
I’ve seen a mountain lion/cougar. Not particularly rare, but rarely seen. I live in Colorado, and know a lot of hunters/hikers/campers, but I’m the only one who’s ever seen one.
It was crossing a road between Ward and Estes Park.
Saw an endangered Florida Panther once, all black and silky-sleek with darkness-piercing eyes, just a short glimpse before it darted back into thick underbrush along the side of a semi-busy South Florida roadway. It was smaller than I thought – perhaps not a full-grown adult. One of the greatest sights of my entire life, especially as a cat lover.