Only Ones

Using any or all of the operations of addition, subtraction, multiplication, division and exponentiation (and brackets), and as many of the number 1 as you need, produce each of the numbers from 2 to 20. For example, here is one way to make the number 17:

$(1+1+1)^{(1+1)} \times (1+1) -1=17$

What is the smallest of 1’s needed to make each number from 2 to 20?

(Note you can’t concatenate the 1’s to make numbers like 11 – each 1 must stand alone as its own number.)

Source