Being a computer scientist/mathematician who has taken just enough physics
to sound knowledgeable, but not quite enough to know better, makes me feel
ready to take a stab at this particular problem. First of all, we should
establish whether this is even an answerable query - is the universe discrete
enough to make this question meaningful?

From an intuitive perspective, it would seem that the universe is about as
continuous as continuous can get. We have rippling water, turbulent winds,
and pulsing suns filled with complex nuclear-powered electro-magneto-dynamic
fluid flow: These are the things that calculus was invented for! Looking
more closely, however, things become less clear. Back in the day a guy named
Max Planck found out that energy only came packaged in integer multiples of
a particular constant (now known as Planck's Constant). This implies that,
in a very strong sense, energy is a discrete quantity. After letting this idea
settle for a while, Planck (and others) tried to find out if there was a
fundamentally indivisible unit for space or time. For time it is a little less
clear, but since mass is energy, mass is pretty clearly quantized as
well. If we take it as a given that there is a fundamentally shortest time,
and assume that that time is around 10^{-128}, then
we've managed to quantify all of time, mass, and energy. This leaves only
position, which is commonly believed to be viewable as discrete in units of the
Planck Length.^{1}

If we take that theory to be true, then for a universe of a given size there
certainly is a maximum number of states. There is a finite number of locations
and a finite number of particles. So now let's try and figure it all out!
Please note that the universe is expanding, so any numbers that come out of
this calculation, if they can be trusted at all, can only be trusted for the
next thousand millenia or so. Also note that the numbers will be
mind-bogglingly huge. You'll have to use all of the techniques you know for
visualizing large numbers just to try and get a handle how big these numbers
really are.

The calculation, when initially stated, seems simple. Remembering that one of
the tenets of modern physics is that two fundamental particles of the same type
are completely the same, we simply take all distinguishable fundamamental
particles and put them in their own buckets, then we will choose
non-overlapping locations for all the particles from the first bucket, then for
the particles in the next bucket, and so on. Unfortunately for us, exactly
what constitutes a fundamental particle, and how many of each kind there are is
currently a contentious issue. People have gathered data on the subject, but
the final word is most definitely not in yet.^{2}

So, in the absence of definitive evidence, we'll go with some rough estimates
and hunches. Searching around the internet, we find that the diameter of the
universe is currently thought to be at least 20,000,000,000 light years,
which turns out to be:

`2*10^10 ly * 9.4605284*10^15 meters/ly *
1/1.6160*10^-35 meters/planck length = 1.2*10^197 planck
lengths`

in the diameter of our universe. Now we'll use the
formula for the volume of a sphere when given the diameter and derive that
there are

`4/3*pi*(1/2*d)^3 = 8.4*10^591`

possible positions for these particles.

Whoah. Already these numbers are pretty freaking huge. Now let's count particles.

Accoring to The Standard Model there are 6 quarks (building blocks for protons and neutrons), 6
leptons (things like electrons), their corresponding antiparticles, and
various force-carrying particles (like photons). Also, other sources have
estimated that there are 10^87 particles in the universe. They are probably
using particle to mean atom, but it's the best number I could find. So we'll
use that number and just have to remember that we're lowballing this
particular estimate. We'll then assume an equal distribution among all types
of particles, which is not right, but will only serve to make our end number
larger, which we've already established it should be. Now, doing that bucket
stuff I mentioned, and not worrying about superposition because the number of
particles, while large, are extremely small with respect to the number of
places to put them, we find that there are (much math elided) 10^24425 ways
of arranging all the particles in the universe. Wonderfully enough, because
the force-carrying particles are included in there, we don't have to worry
about the energy contained in that system, because it is all a property of
quark position! So, at this instant in time, there are 10^24425 possible
states the universe could be in. Which is a whole freaking lot.

Now some of these configurations are more probable than others, but there is still a number of such mind boggling hugeness there that it might make you
appreciate the order lurking in the background that allows you to be a part of
the configuration in which you finish reading this node right...

- Proving this true would get you a free trip to Stockholm.
- Figuring this all out would get you a Nobel Prize as well.
- I forgot velocity! SHIT! Multiply every number by a hojillion to get the right answer.

...now.