Someone on io9 estimated there were about 10^80 electrons in the universe, but I want to ask the Stack Exchange physics community.
This is a well known and kenned estimate. The Wikipedia article Observable Universe covers this well; look especially at the sections under headings "Mass of ordinary matter" and "Matter content — number of atoms" for how it is derived.
The discrepancy between the observed and theorized Galactic Rotation Curves let us estimate what proportion of the density in 1. is made up of dark and ordinary matter. Therefore we now have an estimate for the mean "ordinary matter density" of the observable universe.
Various estimates from spectroscopy and other astronomical techniques give us an inkling of the relative abundances of the elements in the observable universe.
Therefore, given a calculated size of the observable universe, the mean ordinary matter density derived from Hubble's law and extrapolated relative abundances, we can get the number of atoms. This of course gives us the number of protons, therefore the number of electrons.
This estimate does not take into account the possibilities of electrons being paired up with positrons instead of protons to yield a nett charge of nought for the Universe.
It is important to take heed that all this applies to the number in the observable Universe, which may well be much (even infinitely) smaller than the whole Universe.
Adding to and may very well be a subset of @WetSavannaAnimal aka Rod Vance's answer, It is not possible to calculate the total number of electrons in this universe because this universe is local (observable) and global is still unknown, which suggests conservation laws for mass, charge or energy does not apply in this universe.