Entropy does increase in the universe. For systems in which long-range attractive forces are not important (e.g., when gravity is not important), greater entropy tends to be configurations that are statistically more homogeneous (e.g., a uniform gas). But when gravity is important, systems with greater clumping tend to have more entropy. Thus when gravitational systems evolve toward greater entropy, they tend to clump, giving structures like galaxies and stars. For a fixed finite energy in a fixed finite volume; if the energy or volume is unlimited, there is no maximum to the entropy. Entropy would only be maximized if almost all of the energy formed a black hole, the ultimate gravitational clumping.
In our universe, gravitational processes are not fast enough to convert all of the energy into a big black hole, so both entropy and the organization of clumping go up with time. The clumping that formed stars led to nuclear burning in stars and the release of energy in sunlight that led to all sorts of complexity to develop on earth, all the while continuing to increase the entropy of the universe.
If the universe keeps expanding, then there is no limit to the volume, and hence no limit to the entropy, so entropy can keep increasing forever, and the universe is likely never to reach thermal equilibrium (though it might get close if the present acceleration were really due to an absolutely constant cosmological constant, so that the universe does expand exponentially forever, getting closer and closer to what is called the very-nearly-empty thermal state of deSitter spacetime, though theorists have noted problems with assuming that even that state would last forever, suggesting that even all of four-dimensional spacetime as we know it might decay away, but only after a very, very long time).
On the other hand, if the universe recollapses, that is likely to happen too quickly for entropy to reach a maximum then either. (It might go to infinity as the universe recollapses, because in some sense the compression can make the "energy" go to infinity---this is an "energy" that is an energy density multiplied by the volume, not including negative gravitational binding energy that would make the total energy of a closed universe actually equal to zero for all time.)
Thus the clumping tendency of gravity is very important for producing the complexity and structure that has developed within the universe, all the while following the second law of thermodynamics that says entropy increases (or at least does not decrease) and which for nongravitational systems alone (like gas in a box) would just lead to boring statistical uniformity.
Roger Penrose has written a lot about the unusual thermodynamic aspects of gravity that are crucial for our existence, in both his technical and popular books, so you might want to consult them for further details. I'm fairly sure Paul Davies has also discussed this in some of his books.