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...

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