Emma, Abi and Holly from Moorfield Junior School sent us this solution:
The largest number of shapes we got was $22$. We got this by starting with the second square along on the top row. Then we went :
and finished at the first square on the bottom row.
Thomas at Newberries Primary School sent us a picture of the same path through the network:
I don't think it's possible to beat, but maybe you can prove me wrong!