Emirati (UAE)
Blackfoot
Pawnee
Cheyenne
Crow
Apache
Arapahoe
Maori
Eskimo
Inuit
Cherokee
Navajo
Aztec
Utard
And it looks like the OP needs to be amended, since I’d say there are more nationalities (or demonyms) ending with -i than with -ese, plus -er is pretty common.
Nevermind
Official US State demonyms:
Nutmegger (Connecticut)
Hoosier (Indiana)
Down Easter (Maine)
Marylander
New Yorker
Wisconsinite
Wyomingite
Hadn’t run into that one. Why does it remind me of Illuminati? 
Some more demonyms:
Azeri (already mentioned by the alternate of Azerbaijani)
Catalan
Chuvash
Czech
Finn (=Suomi)
Galego (=Gallego)
Gheg
Kirundi
Kurd
Kyrgyz
Lapp (or Saami, if you prefer)
Mordvin
Nepali
Permyak
Pict
Pilipino (= Filipino)
Qaraqalpaq
Tajik (Tadzhik)
Tatar
Tosk
Uighur
Uzbek
Vlach
Not Acceptable, under the OP:
Slovak = Slovakian
Magyar = Hungarian
Sard = Sardinian
A denizen of Nirvana?
Non-standard. The standard form is “Kiwi.” 
Chechen?
Breton?
They do have “-n” tho’ not “-ian” or “-an.”
Lao (also Laotian I guess)
Swede & Finn are noun stems of Swedish & Finnish.
Pole (archaically Polack, which is interesting) is a noun stem for Polish.
Of course there are lots of “-i” forms from the Middle East.
Also Quebecois, or Quebecker, (but not Quebecquese, Quebecan, Quebecki, nor Quebesche).
“-i” seems to be so common (Iraq - Iraqi, Nepal - Nepali) that we should just add it to the OP as one of the standard identifiers.
snarf
Dane?
Oh, jeez, there are a lot of 'em:
Buckeye (Ohio)
Hoopie (not official, but West Virginia)
Keystone Stater (Pennsylvania)
Washingtonian (Washington State)
Peach (Georgia)
Roma, or Romany (Gypsies)
Are there any nationalities ending in “gry”?
(sorry…couldn’t help myself :p)
It’s curious that in America “Polack” is almost an insult, but the Polish word for a Pole is “Polak”
I’m Adyghe.
Lett = An inhabitant of Latvia.
As in “Lithuanians and Letts do it” (Cole Porter).
Wouldn’t that be someone from the obscure Southwest Asian nation of Kiw? 
Saxon.
Spaniard!
Suomi means Finland, Finn would be suomalainen.