> >or, more concisely, (1+1+1)^(1+1) + 1.
> >Can anyone represent that number in fewer than (1+1+1)! ones?

This all depends on what operators and notations are accepted and 
without specifying that, the whole question is useless.

What about without any ones at all: (With C++ operators)
((0++)++)*(((((0++)++)++)++)++)

or without any numbers: (With normal algebra)

((a/a)+(a/a))*((a/a)+(a/a)+(a/a)+(a/a)+(a/a)), a [belongs to] N


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