The 1904 Panama 2.5 centavos ("Panama Pill") is just 10mm. I think it's one of the smallest machine struck coins. Certainly not the lightest at 1.25g though.
Not sure where exactly the "modern" cutoff is, but the 1797 "cartwheel" twopence comes in at 41mm and 56.7g. Chronologically later (mid 1800s), China, Vietnam, and Japan all made massive cast coins well over 40mm. I can't think of any other modern circulation coin larger than the "dollar" which usually does not exceed 39mm.