Well, there's two separate things here.
The example die is in fact three separate words. What were dien and dee in Middle English, became different words spelled die in Modern English. What was spelled deie still has a separate spelling dye. They're homophones, and in the case of the two words spelled die, homonyms; two separate words with the same spelling and pronunciation.
When we come to senses, the difficulty is in how we define the boundary between senses.
(The capacity for words [and symbols, signs, gestures and other units of communication] to have multiple meanings, is called polysemy. Ironically polysemy appears to be monosemic.)
Many senses are very close to each other, and the lexicographer must decide whether they are truly separate enough to count as senses. Figurative uses becomes "full" senses if used heavily, but at what point does this happen? Does an inventive figurative use invent a new sense immediately?
If I've an idiosyncratic use for a word, should you count that as a sense? If a sense listed in the dictionary is marked as archaic should you still count it? What about dialect or colloquial?
Do we count the available productions? Just about any noun can be used in an adjective way, is that another sense? If not, is it another sense if it's a particularly common use of that word?
Also, which words do we count, and do we weigh the results in some way? It would be very easy to produce an answer of "just over 1", by taking every technical word with a limited use such as alpha-dimethyl-cyclohexylethylamine, alphamethyl-phenethylamine and so on, which tend to be monosemic. (And to have several other names, most of which are also monosemic).
Conversely, taking the most common 121 nouns and 70 verbs and counting senses according to the Wordnet dataset in this study came up with an answer of 7.8 senses per noun, and 12.0 per verb.
So that's one answer, though the above should make it clear that we could come up with answers that were very different to those.