Time dilation at the Big Bang At the time the Big Bang happened the matter had enormous density. According the GR (I may be wrong here) such density dilates time.
If so, could it be that the time periods just after Big Bang which are usually considered happening in small part of a second (such as the Planck epoch), in reaity took billons of year (or may be, infinity) but due to time dilation appear to us as spanning only microscopic parts of a second? Could it be that the age of the universe is dramatically underestimated?
 A: Time is a tricky concept in GR because in general different observers will measure different elapsed times. However there is a well defined property called proper time that is invarient and has the same value for all observers. The proper time is normally represented by the symbol $\tau$, and it's defined as:
$$ d\tau^2 = g_{ab}\,dx^adx^b $$
where $g$ is the metric of the spacetime. In the particular case of the FLRW spacetime that (approximately) describes our universe this simplifies to the expression given by mastrock:
$$ d\tau^2 =dt^2-a^2(t)\,d\vec{x}^2 $$
where $dt$ is the movement in time and $d\vec{x}$ is the movement in space (the coordinates in use here are comoving coordinates).
The point of all this is that suppose you choose an observer that is not moving in space, i.e. $d\vec{x} = 0$, then the expression for the proper time becomes:
$$ d\tau^2 =dt^2$$
so the proper time is just the time measured on whatever clock the observer has to hand. So if you are such an observer then the proper time from the Big Bang to the current day is just the time measured on your clock. Furthermore the FLRW metric assumes spacetime is homogeneous, so all stationary (i.e. comoving) observers measure the same time on their clocks - the time since the Big Bang is the same everywhere.
And now we can answer your question. Time wasn't dilated near the Big Bang when the density was higher because (proper) time is by definition what is shown on a comoving observer's clock. Every comoving observer everywhere would have recorded 13.7 billion years since the Big Bang.
A: If one assumes FRW geometry (homogeneous and isotropic universe) in GR, 
$$d\tau^2 =dt^2-a^2(t)d\vec{x}^2 $$
the age of the universe is precisely calculated by (derived from equations of motion and assume that $a(t=0)=0$ is the beginning of our universe)
$$t_0 = \int_0^1\frac{dy}{yH_0\sqrt{\Omega_{\Lambda}+\Omega_My^{-3} +\Omega_R y^{-4}}} $$
where $y=\frac{a(t)}{a_0}$, 
$H_0$:  the present Hubble parameter $\frac{\dot{a}_0}{a_0}$
$\Omega_{\Lambda}$: percentage of dark energy 
$\Omega_{M}$: percentage of matter, including known matter and dark matter 
$\Omega_{R}$: percentage of radiation
*According to Planck 2013 results. XVI. Cosmological parameters, the lifetime of our universe is around 13.8 billion years
*See Cosmology by Steven Weinberg Ch.1 for references, one can also derive this from Friedmann equations
What you have concerned have been taken into account.
Of course, there are many ways to define the concept of time, such as conformal time, comoving time. However, there is only one physical time in each frame of reference, which is the comoving time defined by the frame of the observer (in FRW metric, which is different from the black hole case), 
$$ dt = d\tau \quad \mbox{since} \quad d\vec{x}=0 \quad \mbox{for the observer} $$
One can calculate the conformal lifetime of the universe as well but it is confusing the audience since it is different from measurable time. The comoving time is exactly the meaning of "time" by scientists and even ordinary people. Thus it is the most reasonable way to report the lifetime of the universe in comoving time. 
There are also other details about your question in cosmology, such as the accuracy of GR at the beginning of the universe, how the beginning of the universe is defined. 
(*Note that the speed of light $c$ is taken to be 1 in the above discussion)
A: We don't know if ∞ is the actual age of the Universe or its size. If we presume it is, we have to assume too that Time dilation itself was ∞ (and we wouldn't be here). But we can assume →∞ for each, e.g. a hyperbolic "history" & "size" of the Universe, which make age & size, for all intents & purposes, ∞-1.
A: Is the universe finite or infinite?
If the big bang had infinite mass at the start then time dilation due to gravity would be infinite and time would stand still.  That's the problem with trying to conceive of the start of time because it took infinitely long to start.
An infinitesimal length of time under infinite gravity is infinitely long.
If the universe was finite then time dilation would have some maximum value, giving time a starting point.
