Either Peacocks or Gray Whales, depending on your definitions. I knew the whales were probably going to be there, though it wasn’t the primary reason for the trip; and the peacocks are/were at least semi-wild, though I think they hung around a garbage dump. I don’t know if anyone actually owned them, but they roamed quite a ways away.
Oh, I should elaborate:
Right Whales (was at a restaurant in Kalk Bay harbour when they appeared just outside the windo), Great White sharks (I was just swimming!), elephants (hiking in Knysna forest, years ago), rhinos (apparently escaped a nearby park), hippos (more times than I’d like), crocodiles (same as hippos),cobras (while hiking), giraffes (OK, this was again escaped from a park), naked mole rats, other mole rats, clawless otters, baboons, monkeys, penguins (all while hiking)
Cuttlefish: Was chasing hermit crabs in the shallows in Langebaan Lagoon and whoosh, there they were. Flashing colours and everything.
Oh, we’re native to NYS, all right, born, bred, raised.
Legitimate curiosity: are these rare? The neighbors’ cat used to leave them on our doorstep as presents when I was a little kid, how upsetting if she was actually killing something that didn’t need the help 
Shortly after moving into our house a few years ago (in Michigan, USA), I spotted this phasmid on our screen door. I had lived in Michigan for seven years prior, and never seen one. Haven’t seen one since.
I don’t know what exactly"most exotic" means, but my favorite sightings ever were the basilisks that lived on the grounds of our hotel in Costa Rica.
My backyard is bordered by a small patch of woods, so we get frequented by some interesting animals. There’s the fox, the groundhog (bane of the garden!), occasional deer, a possible coyote (once. But not for twenty minutes), flying squirrels, hawks of some sort, a mother duck and her ducklings crossing the driveway as I was leaving for synagogue. I saw a skunk, too, when we moved in many years ago.
Oh, and apparently there’s a “drunk” squirrel who tips his head back while eating acorns until he falls over backwards.
Heh, that’s what I was thinking - someone upthread mentioned a porkupine as “exotic”, and here, they are a pest and a nuisance. 
Close encounter with an alligator that was sunning itself on the trail in a wildlife preserve.
Scuba diving and getting a little to close for my comfort to a moray. Also swimming in the middle of a humongous school of baby squids.
George Bush Park in Houston has a huge, natural area full of all sorts of critters. It’s also a favored place for placing geocaches. While going after an elusive cache on 03/26/06, I flushed an enormous wild hog. I nearly soiled me skivvies when he busted through the brush. Fortunately, he was going away from me, not at me.
My (then) two year old son found one at our cottage; I was seriously impressed. They are hard to spot, for obvious reasons …
I have 2 (male and female, I guess) that frequent the woods behind my house. I usually see them a lot this time of year, and they are often seen at the same time. Even my wife, who isn’t really interested in birds, finds them impressive.
Maybe the rarest animals I’ve ever seen in the wild were Trumpeter Swans (1989), and Bobwhite Quail.
I’ve seen many Canadian animals others would consider “exotic”- for example, beavers, moose and bears - but they are pretty common here, if you travel through the woods.
The thing I found most exotic to me was a sea creature I found under a rock on the coast of a small Island off Malaysia.
I have no picture, but what it looked like was this: a large, semi-translucent underwater centepede, with many legs and an irridescent skin; it had large, poisionous-looking fangs on the front. About a foot long or so.
I prodded it into a bucket with a stick (which it snapped at), and showed it to my wife. She agreed that it looked thoroughly dangerous, and beautiful in a horrible way. I then let it go.
The reason I call it exotic is I really had no clue at all what the hell it was.
While walking down the railroad tracks in New Jersey, I was astonished by a deer. We just stood there looking at each other for a couple of minutes before it took off.
I called 911 because I was afraid it would get hit by a car. The guy told me they’d had several calls about it.
Used to have 2 mating pairs of painted bunting at our house. Not necessarily exotic but very hard to see because they tend to spend their time in thick foliage. On a college field trip to Arizona I got to see a Western diamondback rattlesnake. Again, not exotic, but for a kid who spent his whole life in the midwest, it was quite a treat. On that same trip, some of the other students spotted and chased a Coatimundi.
Oh, and an animal that really surprised me as being seriously out of its range was an opossum crossing the street in Toronto. Never seen one before (or since).
I was so surprised i looked it up in wikipedia, to find this:
Hm. The rarest would probably be an endangered sand cricket that my mother and I saw digging a burrow at the Great Sand Dunes in Colorado. Another endangered species I’ve seen up close was a Texas Horned Lizard, which somehow got into our kitchen. I scooped him up into a coffee cup and let him go outside. He was cute.
I’ve seen some rare birds in the wild, though very few up close; birds tend to be pretty skittish. I did once see a migrating flight of whooping cranes over our house; they were up so high I needed binoculars to see them, but their call was so loud and unmistakable that we could still hear it easily.
Not particularly exotic to locals, but I was excited to see emus and a wedge-taled eagle in Western Australia. The eagle was huge!
Here in the US, I guess maybe pikas in Sequoia NP. And, probably more “ewwwww!” than exotic, sea hares in the tidepools at Cabrillo NM.
I was playing golf in Scottsdale several years ago. As we approached one of the greens, we noticed that it was covered with jack-rabbits, probably 20-30. That’s not the unusual part, as they are very common. As I was chipping onto to the green, a bobcat rushed out from some nearby underbrush onto the green grabbed one of the jackrabbits and then darted off into the brush.
I don’t know how exotic that is, but it did have an affect on the rest of my game.
Hiking/birdwatching deep in the jungles of eastern Panama (40 miles of uninhabited jungle in every direction) we stopped when we heard a deep coughing sound from around a bend in the trail. It sounded large and very much like a jaguar. Our crazy Panamanian guide and us 4 stupid Americans crept around the corner to see the jaguar (instead of, you know, hiding). Turned out to be a Tayra, a member of the mustelid family, ie, pretty much a big weasel. It was very cool, all black with a long furry tail. He hissed/coughed at us a few times and climbed away through the trees. Pretty exotic in that neither I or anybody I’ve talked to has ever even heard of a a Tayra.
We also saw a Tamandua, an arboreal anteater, on the trip. He was tan and white and very furry with a long pointy nose. Also unknown before seeing one.
I’m going to alter the OPs premise a bit and tell about an unusual chance encounter. The animal, a mountain lion, isn’t really exotic and where my brief encounter took place wasn’t in the wilds, but it was a very unexpected. We were conducting a property boundary survey in the foothills above Glendale, CA and were about a mile and a half west of the Rose Bowl. It being the foothills, the area is rather hit and miss with regards to which lots have been developed and which are in a more natural state. The truck was parked on the street at the end of a long narrow driveway with vegetation on both sides of the driveway. We were putting away the equipment when I just happened to look up the driveway as a juvenile mountain lion crossed it no more than 30 feet from me. I was so stunned I didn’t even react fast enough in order to point the lion out to my associate.
I have regularly seen deer, coyote, black bear, bobcats and even the rare badger while working, but those were all in more remote areas here in Southern California and not entirely unexpected. The mountain lion in the suburbs on the other hand was something unique to me.
I consider myself fortunate that I have seen a good share of the animals listed.
Undoubtedly the rarest I have seen, and one I remember the most clearly, was california condors at the Grand Canyon. We were staying at a lodge on the S Rim, and one morning I walked to the next lodge over to buy some coffee. Near the rim of the canyon right outside of the lodge there were a number of large black birds with numbered tags. There were so many of them - at least 10 - and they were so close, I didn’t really grok that they could be condors. Figured they had to be some species of common vulture. But I quickly ascertained what they were, and woke up my wife and kids to see them.
As there were only 300 or so of the birds at the time (and only half that in the wild), I must have seen something like 5% of the entire population of condors at the same time.
Who knows - maybe they hang out there quite often, and are considered as ubiquitous as pigeons in downtown Chicago, but it realy made an impression on me. And I didn’t see them the next couple of days.
Speaking of pigeons, on the same trip we stayed at a hokey dude ranch type of place W of Vegas near Red Rocks. Old Nevada or somesuch. They had a little zoo, with a cage containing specimens of the exotic species - rock doves! 