The largest Roman numeral which will fit in a Tweet is an "M" plus 279 overbars, which expresses exactly 10^840, as I shall now demonstrate
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The smallest integer with a 281-character Roman numeral which I can find is 78,888,888,888,887,888,888. So I believe the answer is 78,888,888,888,887,888,887, whose Roman numeral is 280 characters long (see next Tweet)
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L̅̅̅̅̅̅X̅̅̅̅̅̅X̅̅̅̅̅̅V̅̅̅̅̅̅M̅̅̅̅̅M̅̅̅̅̅M̅̅̅̅̅D̅̅̅̅̅C̅̅̅̅̅C̅̅̅̅̅C̅̅̅̅̅L̅̅̅̅̅X̅̅̅̅̅X̅̅̅̅̅X̅̅̅̅̅V̅̅̅̅̅M̅̅̅̅M̅̅̅̅M̅̅̅̅D̅̅̅̅C̅̅̅̅C̅̅̅̅C̅̅̅̅L̅̅̅̅X̅̅̅̅X̅̅̅̅X̅̅̅̅V̅̅̅̅M̅̅̅M̅̅̅M̅̅̅D̅̅̅C̅̅̅C̅̅̅C̅̅̅L̅̅̅X̅̅̅X̅̅̅X̅̅̅V̅̅̅M̅̅M̅̅M̅̅D̅̅C̅̅C̅̅C̅̅L̅̅X̅̅X̅̅X̅̅V̅̅M̅M̅D̅C̅C̅C̅L̅X̅X̅X̅V̅MMMDCCCLXXXVII
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