Why are there two Rs in "arrhythmic"?

Solution 1:

Actually, "arhythmic" is recognized by many dictionaries as an alternate spelling (for example, Merriam Webster). As Henry notes, the Greek word is ἄρρυθμος "arrhythmos", so the spelling with one r does not come directly from Greek; it instead seems to derive from a re-combination in English of the elements "a-" and "rhythmic," exactly the way you intuitively wanted to spell the word.

The pattern underlying the main spelling, "arrhythmic," is that words starting with the letter rho (ρ) in Greek, transliterated in this position as "rh," tended to have the rho doubled when a prefix was added or when they were the second half of a compound word. Double rho (ρρ) is transliterated as "rrh." You can find more details and explanation on John Wells's phonetic blog: rh and rrh.

There are many Greek-based prefixes used in English. In general, after any of these prefixes ending with a vowel, a word starting with "rh" will have the spelling changed to "rrh."

Examples of doubing after other Greek prefixes: hyporrhythmic, Antirrhinum, Metarrhizium

For a word that already existed in Ancient Greek, this doubling exists in the original Greek spelling, and is generally brought over to the English spelling. Words newly coined in English from Greek components may follow this doubling rule, or they may not, depending on the intuitions of the person coining the word. For some words, both variants are used, as shown by the coexistence of "arrhythmic" and "arhythmic."

(Incidentally, another word that appears to vary in this way is Metarrhizium; while Merriam Webster gives the doubled spelling I was looking for to use as an example, you'll find that the Wikipedia article calls the genus Metarhizium with only one r.)

Solution 2:

Although it originates in Greek as ἄρρυθμος, this kind of pattern is not uncommon in English.

A comparable example is irregular, which comes from Latin.