Lately my ears started catching words ending in り that seem like nouns created from verbs. I'm sorry I don't have any other examples besides the one from the title 始まる (to start) -> 始まり (the start). Is it just a small number of word pairs like that or there is some magic grammar formula to produce such nouns? ありがとうございます！
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As a rule, a verb's 連用形 (conjunctive/continuative form) can become a noun (名詞化). I think that technically it doesn't matter what word it is. All can take that form and become nouns. In regular use, though, I think you'll find that words that are used this way are relatively limited. So we have common words like 始まり、綴り、しゃべり、 etc. It may be useful to think of these as distinct words that happen to follow this rule of nominalization, because there are quite a lot of words that you wouldn't normally use this way (though I don't think that means you can't). It's just a matter of what's most natural. For example, you wouldn't say 彼の言いを聞いてください. Rather you'd say 彼の言うことを聞いてください, or something along those lines that uses other forms of nominalization.
In my experience, outside of the 'usual' words that you can use, compound verbs seem to be the most common targets of this form, like 話し合い or 打ち合わせ or もの探し. You can also use these verb stems as suffixes. For example you can add ～作り to the end of something to refer to the act of making something (ケーキ作り, or making cake).
There are some words that take on unique meanings in this construction, though, so be wary. Like 見合い, which refers to arranged marriage rather than the literal act of looking at each other.