You can do large multiplications with a convolution and do the carry afterwards.
A convolution can be done with FFT, pointwise multiply, inverse FFT which is O(n log n) rather that O(n^2) for traditional multiplication.
The bits in each limb can be quite small though as there are lots of carries and it depends on how many digits you have and how accurate your floating point is.
Some kind of FFT is how all large multiplications are done.
I had a lot of fun learning about this in relation to GIMPS (the Great Internet Mersenne Prime Search) where you use a variant FFT called a DWT over an irrational base which gives you a free mod 2^n-1 which is what you want for primality testing Mersenne prime candidates using the Lucas test.
