Are there any other words that are spelled with "a" but pronounced with the standard phoneme /e/ in Received Pronunciation like many, any?
Exceptions: derivatives of any (anything, anyone, anytime, etc.); other regional/dialectal mergers (such as the Mary-marry-merry merger); /eər/ as in Mary.
Some people (including me) have /ɛ/ in catch (this is listed as the second pronunciation by Merriam-Webster) although for me the vowel in this word is more variable than the one in many and any (I might say /kætʃ/, while I would never say /æni/).
The past tense of eat, which is standardly spelled "ate," may be pronounced /ɛt/ (see this map from the Cambridge Online Survey of World Englishes). This is only present in the accents of certain regions; I think it is more common in the UK than in North America.
Some people pronounce pall-mall as /pɛlmɛl/ (American Heritage Dictionary) although alternate spellings that fit this pronunciation better do exist, such as "paille-maille" and "pell-mell."
Place names often have unintuitive spellings; one place name spelled with "a" but pronounced with /ɛ/ is the River Thames /tɛmz/ (Oxford Dictionaries).
Loanwords also are often spelled according to different patterns. The German word Doppelgänger is often used in English with the spelling doppelganger; even when the umlaut on the a is dropped, the vowel may still be pronounced as /ɛ/.
