What are some complicated ways to write 1+1?

The standard way when using ordinal arithmetic is: Take the ordinal 1, which is {{}}. Replace each element with a ordered pair of the form {{a},{a,b}} with second element being 0 (that is {}). Repeat with second element 1. Take a union. Take find the ordinal with this order. Overall: otp({ {{{}},{{},{}}}, {{{}},{{},{{}}}} }) Or simplified

otp({ {{{}}}, {{{}},{{},{{}}}} })