How much has the universe expanded in total from the onset of inflation until today? Starting from the onset of inflation, through inflation, reheating, radiation domination, matter domination, the transition toward dark energy domination all the way to today, how many e-folds of expansion have occurred in total?
Obviously there is much uncertainty, depending on the model of inflation, our hazy understanding of reheating, etc. So a rough estimate is fine. Any set of reasonable assumptions will suffice. I'm just trying to get an idea.
 A: Roughly by a factor of $\mathbf{10^{52}}$, or 120 e-foldings, with roughly half of these powers of ten being during inflation, and the other half being after inflation.
According to the latest Planck Collaboration et al. (2016) results, the transition from a radiation-dominated to a matter-dominated Universe took place at a redshift of $z_\mathrm{eq}\sim3400$ (the exact value depends on which data you include; for the "TT+lowP+lensing" data, the value is $3365\pm44$), meaning that the scale factor at this time was $a_\mathrm{eq} = 1/(1+z_\mathrm{eq}) \simeq 0.003$.
The age of the Universe when this happened was $t_\mathrm{eq} \sim 52\,000$ years.
In the radiation-dominated epoch, the expansion goes as $a(t) \propto t^{1/2}$. Inflation is thought to have ended when the Universe was
$t_\mathrm{end}\sim10^{-33}\,\mathrm{s}$
to
$10^{-32}\,\mathrm{s}$. Using a mid-value of
$t_\mathrm{end} = 3\times10^{-33}\,\mathrm{s}$
the scale factor at that time must have been
$$
a_\mathrm{end} \simeq a_\mathrm{eq} \left( \frac{t_\mathrm{end}}{t_\mathrm{eq}} \right)^{1/2}\sim 10^{-26},
$$
so the Universe has expanded by roughly $\ln(10^{26})\sim60$ e-foldings.
Coincidentally, the Universe is thought to have expanded by roughly the same factor during inflation, so in total, since the onset of inflation, the expansion has gone trough roughly 120 e-foldings.
The largest uncertainty of this is probably the duration of the inflation; I think the 60 e-foldings is the minimum needed, but it could in principle be more.
Size of the Universe just after inflation
Today, the radius of the observable Universe (the distance to the particle horizon) is $R_\mathrm{now} = 14.2\,\mathrm{Gpc}$ (or 46.3 billion lightyears). Thus, at the end of inflation, the radius was
$$
R_\mathrm{end} = a_\mathrm{end} R_\mathrm{now} \simeq 5\,\mathrm{m}.
$$
Using $t_\mathrm{end} = 10^{-33}$ or $10^{-32}$ seconds yields a radius of $\sim3$ or $10$ meters, respectively, so it doesn't make a huge difference.
Python script
I used the following script in Python (which requires installation of the package astropy):
import numpy as np
from astropy.cosmology import Planck15
from astropy import units as u
z_eq  = 3365.                           #Redshift at -matter equality (TT+lowP+lensing)
a_eq  = 1 / (1+z_eq)                    #Scale factor at rad-mat eq.
t_eq  = Planck15.age(z_eq).to(u.s)      #Age at rad-mat eq. in sec
t_end = 3e-33 * u.s                     #Age at end of inflation in sec
a_end = a_eq * np.sqrt(t_end/t_eq)      #Scale factor at end of inflation
R_now = Planck15.comoving_distance(1e7) #Distance to particle horizon today;
                                        #  using 1e6, or even 1e5 or 1e4 makes little difference
R_end = a_end * R_now
print('Age at rad-mat eq.:              ', Planck15.age(z_eq).to(u.yr))
print('Scale factor at end of inflation:', a_end)
print('e-foldings since inflation:      ', np.log(1/a_end))
print('Radius of Universe today:        ', R_now.to(u.Glyr))
print('Radius at end of inflation:      ', R_end.to(u.m))

