{ "mybigint" : -123434580239458203948203982345723458 }
You can throw the spec out the window and put as many digits as you want into a JSON integer. Any half decent parser in a half decent language will accumulate the token and spit out an integer object.There is no reason to write a JSON parser that doesn't accept bignum integers, in a language that has them.
There are only two reasons some JSON implementation doesn't accept them. One is that the underlying software doesn't handle them. In that case JSON is moot; it's a limitation of that system. You cannot input bignums into that system through any means; they are not representable. The other is that the JSON implementation was written by obtuse blockheads: bignums could easily be supported since the underlying language has them, but aren't simply for compliance with the JSON specification. Compliance can be taken to counterproductive extremes.