Why is "archaic" pronounced uniquely? Is the sequence -ɪɪ- only found in this word?

Just to be clear. The base Greek Word is αρχη, pronounced archi - but the 'ch' is like a guttural 'h', meaning beginning or origin. The adjectives formed from it are αρχαιος (pronounced archayos) and αρχαyϊκος (with the 'i' separately pronounced). They differ slightly in meaning. The former means ancient or old; the latter has the overtone of 'out of date', particularly in the sense of the use of an antique form of a language for some sort of literary effect ('archaïsing'). Matters are all the trickier because the mediaeval scholars had some pretty strange ways of pronouncing the ancient Greek, as did I as a school boy. The authentic pronunciation of the original words (which differed regionally, just as English does) was only fully established in the latter third of the last century.

This should be a comment, but it wont let me make a comment. In elementary American english grammar rules, a vowel followed by another vowel, or a vowel followed by a consonant by another vowel is a long vowel. pronounce it as three syllables: ar + cha (long a) + ic (short i). which is indicated in the pronunciation representation "-keIIk" spelling (same vowel-vowel rule). The a in the middle syllable is long because it is immediately followed by another vowel.

In English, there's a phoneme commonly called "long A" (because it evolved from what used to be a lengthened /a:/). This part's pretty uncontroversial: it's the phoneme in the middle of "face".

However, linguists have different views on how to transcribe this sound. It's often pronounced as a diphthong, so some people write it as /eɪ/, /ei/, or /ej/; other people just write it as /e/ for simplicity, and say it's fundamentally a single unit. This mostly comes down to a transcription convention. One standard way to talk about it (in a purely English context), without committing to any particular transcription, is "FACE" or "the FACE vowel".

In the word "archaic", the underlying phonemes in question are this "long A" (the FACE vowel) followed by "short I" (the KIT vowel). Some people transcribe this as /eɪ.ɪ/ or the like. But for me, I certainly don't pronounce the same vowel twice in a row: the off-glide of FACE is higher than the vowel in KIT. So I prefer to transcribe it instead as /ejɪ/, to emphasize this height difference.

The same sequence appears in a lot of words, when a suffix starting with KIT is attached to a root ending in FACE: others have mentioned "mosaic" and its ilk, but it also shows up in "laying" and so on.

P.S. As for how it happened: note the diaeresis in the Greek ἀρχαϊκός, indicating two separate vowels /a.i/ with a syllable break between them. When it was eventually borrowed into English (via Latin and French), this was how it was pronounced; later sound changes turned /a/ into /ej/ and /i/ into /ɪ/.