Across 147,478 entries this dictionary records 207,235 distinct senses — an average of 1.4 per entry. Only 825 entries, 0.6% of the corpus, carry ten meanings or more, and almost all of them are short, common, everyday words: break, cut, run, play, make. Rare and technical words carry one meaning because they were coined to carry exactly one.
The shape of the distribution
207,235 senses across 147,478 entries sounds like plenty to go round. It is not distributed evenly. The great majority of entries carry a single sense, and a very small number carry dozens.
825 entries — 0.6% of the corpus — have ten senses or more. At the extreme, “break” carries 75 in this corpus, “cut” 70, “run” 57, “play” 52 and “make” 51. Every one of those is a one-syllable word in the top commonness band.
At the other end, 99,506 entries sit in the rarest band, and most of them mean one thing. A species name, a mineral, a piece of clinical vocabulary: coined for a referent, still attached to it.
Why the commonest words collect the most meanings
New meanings do not attach to words at random. They attach to words that are already available — short, familiar, cheap to say, already in everyone's mouth. When English needs to talk about something new, it reaches first for what it is already holding.
So “run” acquires a sense for machinery, for software, for a candidate in an election, for a length of stocking, for a score in cricket. None of these required a new word; each borrowed a word that was already doing several jobs and could take another.