What are the essential Japanese counter words?
For example -dai for machines, -mai for papers and stuff like that.
- What else to expect/know?
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There are entire dictionaries for this (数え方の辞典). Here's a link to a whole bunch. 個 (ko) is the most commonly used one. Japanese people use it on almost everything, including stuff that has its own counter, partly because it's sometimes annoying for even them to think of the proper 数詞.
Etc. The list goes on. Just learn the most common ones and use 個 for everything else, learning the less common ones as you go.
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The article at Wikipedia covers the common ones as well as a decent number of extended ones, and lists the exceptions for days, people, etc. as well as rendaku and number word changes (e.g. 300->san*bya*ku, 4:00-> *yo*ji).