I’ve got around 10 observations that are firsts on iNaturalist. Pretty easy to find them, particularly South African-endemic aquatic insects.
I’ve got about three observations that I believe are the first photographs ever of living specimens: https://www.inaturalist.org/observations?project_id=71923&user_id=davidklop
I’m not sure if I’ve ever seen a mallard in my life.
Most:
Least:
Fewest global observations:
For most global observations, I have multiple observations of the mallard, western honey bee, and Asian lady beetle, which are the top three species globally, so I assume it’s those!
The mallard is an obvious winner at the most observations but the one with the fewest obs… https://www.inaturalist.org/taxa/448085-Turritopsis-dohrnii the immortal jelly is a winner for me.
For the global stats.
The most commonest species is Black-crowned Night Heron, though I have got only one observation of it.
For the rarest. I have an observation of Rhinyptia indica, which is a iNat first.
PS. I have never seen a Mallard.
For commonest probably Mallard, and for rarest, I have a few iNat onlys.
Hello cydork! Welcome to the iNatForum.
(Love your avatar, btw)
I thought I would mention a useful link that was posted here a few years back for checking the ‘rarity’ (not always reflective of actual rarity of a species, more likely, how difficult it is to get a picture of one!) of any user’s list. And if you’re the super competitive type, there’s even a place where you can compare yourself to any other user.
Here it is: https://elias.pschernig.com/wildflower/leastobserved.html?user=
Just type a user name in the top box and off you go. There’s even filters for observation grade there too.
The ‘most observations’ side has been covered pretty well here. Mallards and Asian ladybugs are like the Google and Facebook of the iNat observation piles!
I’ve got (still!) a only-one-spotted-so-far RG on my list. A kind of freshwater midge (that I had no idea was so rare!). https://www.inaturalist.org/observations/162865392
After that, it’s 7 species with under 10 observations, but only 3 of those are RG.
So, my least observed is Golden Saxifrage: a high-altitude species with limited distribution. I have 2 of 93 total observations; most of the observations are in Colorado, so I expect that mountain tourism skews the distribution a bit.
Most observed? Good ol’ Anas platyrhynchos, but at least I’ve only contributed 13 of them.
On the other hand, the Plains Forktail, Ischnura damula, has 1622 observations total—of which I have 131. Did I mention that I love my little flying neon matchsticks? 
My most observed is the Eurasian Magpie, and one of the rarest is Cryptops hortensis, with around 360 obs worldwide, and I observed three of them :)
It’s nice to know that someone might still have the excitement of posting a Mallard as a lifer! 
how does one know which species they observed has the lists total obs?
I’ve seen the first 10 of the most observed species worldwide - up until Eastern Gray Squirrel (Sciurus carolinensis), which I have never seen.
As of today, there are about 17 species where I am the only observer on iNat, most of which are from Turkey.
These have a special story:
Pyrausta amatalis and Cynaeda escherichi are two small, closely related moths that I found within two days in the same area. As far as I can tell, my photos are the only existing photos of live specimens (at least on the internet), since I could only find some photos of pinned museum specimens.
My personal favorite, however, is this freshly emerged Chamaesphecia moth, which might be an undescribed species, possibly making me the only person to ever see this species at all. The plant I found it on, some kind of catnip Nepeta phyllochlamys, may well be the host plant. In any case, I’m also the only observer on iNat for that plant.
(@insectobserver123 See @broacher’s reply for the handy link.)
Welcome, @cydork !
My most will be the Western honeybee and I have a few rarities as well but that is likely because I hyperfixated on one space for years, not because they are necessarily rare things to observe.
The least observed organism I have observed is the fungus Phyllactinia marissalii, I took a picture of what I thought were insect eggs with my new macro lens and it turned out to be a hemisphere first for this fungus! https://www.inaturalist.org/observations/184020643
The most observed I have is the Western Honey Bee, which is the second most observed species globally (second to Mallard)
I posted an observation today (https://www.inaturalist.org/observations?taxon_id=432909) that turned out to be the first observation of that species … at least, when I go to the species observations page, there’s only one observation listed. I’m gobsmacked, to say the least.
Wow! There is a project for that but I am not sure what the name is. Anyone?
My submission of Collared Peccary, Pecari tajacu, is one of 16057, making it the most diluted of my submissions. https://www.inaturalist.org/observations/7726979
I have the only two (RG) submissions of Wernerius mumai, a small scorpion found in western Arizona. https://www.inaturalist.org/observations?taxon_id=428346&verifiable=any
I also have the only four (RG) submissions of Vaejovis miscionei, another small, recently described Arizona scorpion. https://www.inaturalist.org/observations?taxon_id=1323566&verifiable=any
I also have one of the two Graemeloweus maidu, a scorpion found in the western Sierra Nevada foothills of California. https://www.inaturalist.org/observations/56957820