Is "princessship" a real word? Are there any other words which have the same letter 3 times consecutively?

I believe princessship is typically spelled with a hyphen princess-ship, although I do remember references to it without. However, if your friend is willing to allow princessship as a word, certainly words like dutchessship, governessship, countessship, etc. would also qualify.

So one way or another, you can prove your friend wrong =D

Edit: A quick search also brings up frillless which has an entry in OED

I'm not a native speaker, but I'm under the impression that words with the same letter repeated three times are either elided so that they contain two, or are spelled with a hyphen.

For instance cross-section (hyphenated) or chaffinch (where the 3rd 'f' is omitted)

Discounting acronyms, for words without hyphens that have three consecutive letters, the OED lists these:

brrr, countessship, duchessship, frillless, governessship, grrr, grrrl, hostessship, postmistressship

For words that have a letter, a hyphen, then that same letter twice repeated, it lists these:

eve-eel, paste-eel, salpingo-oöphorectomy, salpingo-oöphoritis, slime-eel, snipe-eel, spine-eel

For words where you have the same letter twice, then the hyphen, then that same letter again, it has all these:

ass-ship, ball-less, ball-lightning, ball-like, bee-eater, bell-less, boss-ship, boss-shot, boss-stone, brass-smith, burgess-ship, call-letter, call-loan, carcass-shell, cell-layer, cell-less, cell-like, cell-lineage, cell-lined, chaff-flower, class-subject, compass-saw, compass-signal, compass-stock, cross-saddle, cross-sea, cross-section, cross-sectional, cross-sectioning, cross-seizing, cross-sell, cross-shed, cross-shoot, cross-shot, cross-sleeper, cross-spall, cross-spider, cross-springer, cross-staff, cross-stitch, cross-stone, cross-street, cuckoo-orchis, cypress-spurge, demigoddess-ship, distress-sale, doll-land, dress-shield, egg-glass, fall-leaf, fall-less, fee-estate, fee-expectant, frill-lizard, fusee-engine, gall-leaf, gall-less, gill-lamella, gill-less, gill-lid, glass-sand, glass-shaped, glass-shell, glass-shrimp, glass-slag, glass-snail, glass-snake, glass-soap, glass-sponge, goddess-ship, grass-seeder, grass-siding, grass-snake, grass-snipe, grass-sparrow, grass-spirit, grass-sponge, grass-spring, head-mistress-ship, ill-less, ill-lived, ill-looking, ill-lookingness, isinglass-stone, Jaycee-ette, joss-stick, kill-lamb, kiss-sky, knee-elbow position, mastiff-fox, mill-lands, mill-leat, mill-lodge, miss-stays, mistress-ship, moss-seat, moss-starch, no-see-em, off-faller, off-falling, off-farm, off-field, pass-shooting, patroness-ship, poss-stick, press-shy, press-stone, press-stud, princess-ship, process-server, puff-fish, quill-less, seamstress-ship, see-er, see-everything, shell-less, shell-lettuce, shell-lime, skull-less, small-leaved, smell-less, stall-literature, still-liquor, stress-strain, three-edged, three-eight, till-land, toll-lodge, troll-line, wall-less, wall-lining, wall-lizard, well-laboured, well-labouring, well-laced, well-laden, well-laid, well-languaged, well-learned, well-leaved, well-led, well-legged, well-lettered, well-lighted, well-liked, well-liking, well-limbed, well-limned, well-lined, well-lit, well-liveried, well-living, well-loaded, well-lodged, well-looked, well-looking, well-loved, well-lunged, will-less, will-lessly, witness-stand, zoo-organic

Alas, there appear to be no instances of four letters in a row, even if separated by a hyphen or an apostrophe, such as *Kwanzaa-aardvark, *frisbee-eel, *tatoo-ooze, *install-llama, or *chimpanzee-eerie.

I see a product opportunity here. :)

How about "Wiiitis"? Although I'm not sure the word is considered a "real word".

Wiiitis -- pronounced “wee-eye-tis” -- is the latest ailment to develop from the video game era https://www.reuters.com/article/us-wii-elbow/if-its-not-tennis-elbow-it-may-be-wiiitis-idUSN0616721120070606