I was taught that ê is pronounced /ɛ/, but the ê in dépêcher seems to be pronounced as /e/. Curiously, the ê in "je me dépêche" clearly sounds as /ɛ/.
Is "dépêcher" an exception of that pronunciation rule? If so, are there more exceptions to this rule in other usual words?
I don't think it's an exception so much as a way of pronouncing as /eXe/ the end of certain words that can also be pronounced as /ɛXe/, where X stands for some consonant sound.
The same pronunciation is observed, for example, for all the verbs in -êter that I've found in wiktionary, such as
Prêter / Apprêter
Enquêter
Arrêter
S'entêter
Étêter
Embêter
In all of these cases, the pronunciation of the end of the word can be /ete/ or /ɛte/; both will be understood just as well. Frequencies of each option may vary between words and regions, but I don't have any data about that. The same flexibility in pronunciation options exists for other conjugations that end in /te/, such as the second person plural indicative present (vous prêtez) or the participe passé (j'ai enquêté)
In all of these cases, /ɛ/ must be used in conjugations that end with /tə/, such as the singular present indicative. Using /e/ would be characterized as a mispronunciation, e.g. to pronounce the ê in il m'embête or j'arrête, one must use /ɛ/, not /e/.
I'm not sure about cases like "nous arrêtons, vous enquêtiez". My sense is that /e/ is OK here too.
Finally, it's not specific to verbs ending in -êter or -êcher, the same possibility of sliding from /ɛXe/ to /eXe/ happens also at the end of words like aimer, baisser, mêler, laisser, errer, etc.
(Thanks Gilles ♦ for helping me see that it is a more general phenomenon than my first response.)
It's part of a more general phenomenon. Standard French makes a phonemic distinction between /e/ and /ɛ/, but in practice there are considerable phonetic variations between [e] and [ɛ], some regional, some depending on the speaker, and some free variations (where the same speaker might randomly pronounce the same phrase in one way or the other). A majority of French speakers (excluding the South-West of France where both are systematically realized as as [e] in open syllables and as [ɛ] in closed syllables) would probably tell you that /e/ and /ɛ/ are clearly different sounds, but the reality says otherwise, and many scholarly articles have been written on the topic.
French has a general rule that /ɛ/ (the semi-open vowel) occurs in closed syllables, i.e. when the vowel is followed by a consonant, and /e/ (the semi-close vowel) occurs in open syllables, i.e. when the syllable ends with the vowel, except that both /e/ and /ɛ/ can occur at the end of a word. There are similar pairings between /ɔ/ and /o/, and between /œ/ and /ø/. For example : pétrifier /pe.tʁi.fje/ vs certifier /cɛʁ.ti.fje/, élève [e.lɛv] vs élevé [e.lə.ve].
This general rule has a lot of exceptions, however. One of the exceptions is a form of assimilation called vowel harmony. Vowel harmony is not a driving principle in French the way it is in some other languages, but it does exist a little, and I think this is the main example. A lot of words in fact contain /ɛ/ in an open syllable, and it is realized as [ɛ] most of the time. But if the following vowel is /e/, then the first vowel is realized as [e] far more often. For example:
/e/ before /ɛ/ normally has no impact. For example prétraitement /pʁe.tʁɛ.t(e.)mɑ̃/ is not pronounced /pʁe.tʁe.t(e.)mɑ̃/ much more often than prétraitement /tʁɛ.t(e.)mɑ̃/ is pronounced /tʁe.t(e.)mɑ̃/, which is to say, rarely (but not never). However, the succession /e/-/ɛ/-/e/ is an exception. The succession /e/-/ɛ/-/e/ where the middle syllable is open is really hard, and the middle syllable is almost always realized with a semi-close vowel. Where /ʁə.pe.ʃe/ vs /ʁə.pɛ.ʃe/ for repêcher is mostly free variation, dépêcher is almost systematically pronounced /de.pe.ʃe/.
This tends to bleed into other tenses of the verb where the harmony rule wouldn't otherwise happen. For example je dépêche is /ʒə.de.pɛʃ/ because the last syllable is closed, but nous dépêchons is usually /nu.de.pɛ.ʃɔ̃/ even though there's no /ɛ/-/e/ succession.
I couldn't find a reference specifically on /e/-/ɛ/-/e/. Here are a few references regarding [ɛ] vs [e] in general.
I don't think it's an exception, arrêter follows the same pattern.
The circumflex accent is more historical (or for breaking ambiguity) than about pronunciation.
To clarify, I said I don't think it's an exception, not because there are other words like it (arrêter is the only one I can think of) but because there is no real rule about pronunciation with a circumflex.
Pronunciation has changed as time went by and the ê in "dépêcher" is really pronounced /e/; however, this word hasn't been the object of the 1990 reform which resulted in "événement" becoming "évènement" precisely because the é in "événement" was not pronounced /e/ anymore but /ɛ/.
The TLFi still carries the form /de pɛ ʃe/ and explains that the form /de pe ʃe/ is used for "vocalisation harmonique", but I never heard anything else than the form /de pe ʃe/.
( reverso) harmonisation
nf (phonétique) fait d'harmoniser, de rendre conforme aux lois des accords en musique, ou de mettre les différentes parties d'un ensemble en accord parfait
(encyclopédie libre) : L'harmonie vocalique est une modification phonétique concernant les voyelles d'un même mot ou syntagme ; il s'agit d'un type d'assimilation à distance (ou dilation) des timbres vocaliques entre eux : les syllabes d'une même unité (comme le mot ou le syntagme) doivent toutes présenter à la suite des voyelles « compatibles », c'est-à-dire appartenant à la même « classe » que celle de la voyelle précédente, laquelle classe varie selon les langues.
The pronunciation given here, Larousse en ligne, is only /de pe ʃe/ (you can listen to it).
