How fast was the universe expanding during the first second of the big bang?
I don’t think it is possible to answer this.
Suppose you had a metre stick and a stopwatch, you could measure the speed of an object by dividing distance by time, right? (Admittedly you may need a large stick and a super-dooper stopwatch, but in principle it can be done.) This doesn’t work for the expansion of the universe, because as space expands so will your metre stick.
Instead, cosmologists refer to expansion in terms of z, the red-shift which is effectively the ratio of distance separating two objects now to the distance separating them at some point in the past. (More precisely a ratio in the wavelengths of emitted light.) Clearly, this approaches infinity as you get closer to the initial singularity we call the Big Bang.
You do, however hit another theoretical limit before then: the universe was opaque to electromagnetic radiation before it cooled sufficiently for electrons to be bound into atoms. Effectively the light was constantly being absorbed and re-emitted so could not travel across the universe. The point at which this ‘settling’ occurred was some 380,000 years after the Big Bang and gave rise to the cosmic microwave background radiation (CMBR) which has a redshift of just over 1000.
Predicted, but never observed, are events such as the release of neutrinos or gravitational waves from the Big Bang at redshifts of over
Not sure this constitutes an effective answer, but it was a very interesting question!