Where is the "pere" in "ammeter"?

The instrument to measure current in amperes is called ammeter, while the instrument to measure charge in coulombs is called coulombmeter.

What happened to the -pere? Is there a historical reason for this?

Ampere-meter stands for the seldom used unit A*m.

Ampère, named after André-Marie Ampère, is first attested in the Oxford English Dictionary in the same year, 1881, as the term was adopted by the Paris Electric Congress. Ammeter appears a year later. In that short time, it rather looks as if the phonetical process of elision became reflected in the word’s spelling. Elision occurs when a sound is lost by the influence of those around it, and a consonant is particularly vulnerable when three occur together. In this case, the bilabial consonant /p/ seems to have been squeezed out by the repeated bilabial consonant /m/ that would otherwise have produced ampmeter.

Most likely, it's derived from the common short form of the unit, amp (pronounced /ˈæmp/), which is obviously derived by shortening ampere. Then, in speech, the /p/ gets elided from the awkward /mpm/ sequence of *ampmeter, giving the current pronunciation.

(I don't know why the p was deleted in spelling, though. Perhaps it was filtered through a language like Italian that conventionally drops silent letters in spelling.)