jmc

The first few terms are $$\begin{array}{rl}{z}_2& =0^2+i=i,\\ z_3& =i^2+i=-1+i,\\ z_4& =(-1+i{)}^2+i=-i,\\ z_5& =(-i{)}^2+i=-1+i.\end{array}$$Since $z_4=z_2,$ and each term depends only on the previous term, the sequence from here on is periodic, with a period of length 2. Hence, $|z_{111}|=|z_3|=|-1+i|=\boxed{\sqrt{2}}.$