At the other end of the spectrum, ‘row’ has two different pronunciations (each with its own meaning).

I do think this is a great point but I do think the idea is fun to consider anyways. My first thought would be some (silent) {super,sub}script symbol/letter to indicate distinction between them. Obviously this feels like we'd be right back where we started but if every non-homophone was neatly standardised (and, as my sibling comment pointed out, every homonym made unambiguous) that seems promising! This is, of course, operating under the assumption that homophones are primarily edge cases (or that the other words in the homophone group are not often used in the same context, or at all).

Minimal research turned up two lists. One only lists groups of n > 2 (88 triples; 24 quadruples; 2 quintuples; 1 sextet; and 1 septet) [0]. The other is for British English (441 groups) [1].

Your example, by the way, is a particularly interesting case! From the Wikipedia page [2]: 17% of Americans (primarily in the Northeast and most clearly in Philadelphia, New York City, and Rhode Island) pronounce each distinctly, with a further 26% merging only 2/3 pronunciations. Accordingly, their distinct IPA pronunciations are /ˈme(ə)ɹi/, /ˈmæɹi/, /ˈmɛɹi/, respective to your ordering (the last, merry, is the one we've converged on). More frustrating still, the list of "multinyms" in [0] excludes this example so it's especially difficult to know how many there may be in practice.

[0] https://web.archive.org/web/20160825095711/http://people.sc....

[1] http://www.singularis.ltd.uk/bifroest/misc/homophones-list.h...

[2] https://en.wikipedia.org/wiki/English-language_vowel_changes...