What is the largest number we can get using $n$ ones, addition, multiplication and brackets?

Since

1) it doesn't make sense to create any 1+1+1+... terms with more than 3 1's, since 1+1+1+1+1 is 'worse' than (1+1) times (1+1+1), and for more 1's it will be even 'better' to multiply several 1+1+.. Terms rather than have one long one, and with 4 1s 1+1+1+1 is just as good as (1+1) times (1+1),

and

2) two 1+1+1 terms is better than three 1+1 terms,

your basic strategy is to get as many 1+1+1 terms as possible. So:

If n mod 3 = 0, the best you can do is $3^{n/3}$

If n mod 3 = 1, get two 1+1 terms, and otherwise 1+1+1 terms, so you get $4*3^{(n-4)/3}$

If n mod 3 = 2, get one 1+1 term in addition to 1+1+1 terms, so $2*3^{(n-2)/3}$