In first-language English pronunciation (Australian, British, American, etc., not Indian, Malaysian, etc.) are there any words with the /e/ (or /ɛ/) sound in "bed" /bed/ at the end of a word? As a counter-example "me" is pronounced /miː/. I don't know of any words ending in "e" where it is pronounced /e/; it is usually either /iː/ or silent. If you do know of any words please specify if they're limited to a certain dialect, region or accent.
No, there cannot be.
Terminal /e/ can only occur as a falling diphthong, as in they. Actually, it is very rare to have /e/ without it being an /ej/ diphthong anywhere in English.
And unstressed /ɛ/ will soon enough go the way of all things, despite what bokeh enthusiasts would have you believe. So it will soon enough be a schwa or gone.
There are some unassimilated loanwords (mostly from French) that have a non-silent e at the end, which usually retain an acute accent to indicate the unusual-for-English pronunciation. For example: blasé, café, cliché, fiancé, flambé, frappé, macramé, olé, protégé, risqué, soufflé, and touché. However, there is a tendency to dipthongize the /e/ to /eɪ/, so this may not count.
Yes, there are.
But first of all, what dialect of English are you talking about? There are many different dialects of English, and answers to pronunciation questions will differ among them. Second, which sound are you actually talking about? There are more than one 'e' sounds you could be referring to, such as /eɪ/ (as in "rein"), /ɛ/ (as in "bed"), and /ə/ (examples differ greatly among dialects).
Examples with /eɪ/
Dan already pointed out a number of examples. Here are some more:
I don't see why these should be disqualified just because they end with /eɪ/ rather than /e/, since a pure /e/ occurs very rarely, if at all, in most dialects of English. I also don't see why they should be disqualified because they are borrowed from French: almost 30% of words in English are from French.
If you consider the same sound but not spelled with the letter 'e', then there are many more examples, including
and so forth.
Examples with /ɛ/
This does occur, although admittedly uncommonly, since English phonotactics tend to discourage final or unstressed /ɛ/. Such words tend to be monosyllabic interjections. Here are some examples:
There are a number of borrowed words that may count, although some may pronounce them with /ə/ or /eɪ/.
Examples with /ə/
There are plenty of such examples, including
Pace Peter Olson, no one's mentioned the
Examples with /ɛ̃/
This is the nasal version of ɛ that, like the click consonants in 'tsk, tsk, tsk,' makes vanishingly few appearances in English.
Here are two examples:
bleh/bleeah/bleah - interjection of disgust/discomfort/indifference http://en.wiktionary.org/wiki/bleh
"'weh weh weh' (or 'wah wah wah') cried the baby" - onomatopoeia, obviously
These are both native English as far as I can tell.
They both rhyme with 'Gaugin' (in my dialect anyway) -- the sound ɛ̃ is part of the regular vowel inventory in French. Educated English speakers seem to invoke it often enough to at least pronounce words like 'Gaugin' half-way correctly.