There are a small number of words aside from ageing that retain silent e before -ing; for some of them, this spelling is mandatory in American English as well as British English. For others, it is optional in both of these varieties of English. Retained silent e before ing seems to occur mainly after vowel letters (including y) or after the letter g, especially in the sequence nge.
Of course an e cannot be dropped when it is used as a single letter to represent a vowel sound directly before the suffix -ing, as in being or the trisyllables segueing, sauteing (or sautéing) and appliqueing (or appliquéing).
The digraph ee also retains both letters before -ing (seeing, freeing, decreeing), even though one e is lost as a rule when -ee comes before a suffix that starts with e such as -ed, -er, -est.
It is standard to drop the letter e that is used after the letter l or r to indicate a syllabic liquid or a sequence of schwa + liquid: rambling, massacring, and in British English centring, manoeuvring.
Two important mandatory examples: dyeing (not dying), singeing (not singing). Dropping the e in these words would cause ambiguity with the -ing forms of die and sing respectively.
Verbs ending in -oe also seem to mandatorily keep the e (-oeing):
This doesn't avoid any lexical ambiguity as far as I can tell, but perhaps helps indicate that they are not pronounced with the diphthong usually represented by oi, as in the onomatopoeia boing. Going and doing exist despite this, however.
- -ie pronounced as /i/ (unstressed): sortieing
This is complicated a bit by the existence of an alternative spelling sorty, which means that sortying is also a valid form. However, I haven't found any source that supports a paradigm like sortie, sorties, sortying. The same applies to stymie, as far as I can tell. At any rate, it's clear that the gerund/present participle form of a verb ending in -ie is generally not spelled with -iing.
This is a rare word meaning "prophesying", from Scots. There are probably other Scots verbs ending in -ae that have some use in English. All of the dictionaries I've consulted that list inflections for this word in modern English show it as retaining the e in its -ing form.
Retaining the "e" is optional for some verbs.
Ending in consonant sounds:
-nge pronounced as /nd͡ʒ/: tingeing, twingeing, bingeing, whingeing, swingeing*
*I think the e of swingeing is usually kept to distinguish it from swinging, the -ing form of swing, but the AHD indicates that the e may be dropped anyway. The word swinge is used more in British English than in American English.
-gue pronounced as /g/: rogueing, vogueing
-ge pronounced as /ʒ/: lugeing
Ending in vowel sounds:
-ie pronounced as /aɪ/: vieing? This looks very odd to me, and no major modern online dictionary lists it, but there are a small number of more-or-less contemporary examples from Google Books and from online sources (for examples, see Vocabulary.com and wordnik): these can probably just be regarded as misspellings, although they cannot be dismissed as typos because there are sources that use this spelling consistently (such as Social Ontology: Recasting Political Philosophy Through a Phenomenology of Whoness, by Michael Eldred).
the extremely rare inoneing, listed in the OED with two citations, from "in-" + "one" or the phrase "in one". Obviously one has a very non-standard pronunciation for a word ending in the letters "one".
"routeing", mentioned by Muzer in a comment:
It's definitely archaic if ever used at all, but for some reason the rail industry in the UK uses "routeing" as the official spelling of eg "The Routeing Guide". I've very rarely seen this spelling outside this industry.
(Also see "Skyping or Skypeing".)
There are also a number of more or less common nonstandard spellings/misspellings with this feature, such as "battleing" (for standard "battling"). The variability in people's intuitions about whether to include a silent "e" here seems similar to to the variability before the suffixes "-able", "-ish", and "-y".
Examples compiled with the help of Onelook wildcard search and tchrist's post in chat.
I think there are better/more complete descriptions in many sources about English; you could probably find some on Google.