If the universe is infinitely large, it could still be growing. If, from any point in the universe, the rest of the universe seems to be moving uniformly away from the observer, then the universe is obviously growing--even if it is infinite in volume!
In fact, we know from redshift that the rest of the universe is moving away from us: stars, nebulae, galaxies, entire clusters of galaxies--all are moving away from us. The farther away they are from us right now, the greater their speed away from us. (See: Hubble constant.)
A commonly used analogy to help explain this is to picture some two-dimensional beings living on the surface of a balloon. The beings, unaware of the curvature of the balloon in three-dimensional space, look in all directions and see an infinite expanse of two-dimensional space. If one of them decides to embark on a journey to "the end of the balloon" by walking in a straight line, he will eventually circle around and end up exactly where he started, without ever having reached a boundary or deviated from his straight-line path. The two-dimensional being is completely within his rights in assuming that his balloon universe is infinite.
Now imagine a careless Everything user comes along and starts blowing more air into the balloon. Pandemonium! Squiggly flat people who were standing right next to each other having a conversation just moments ago are suddenly across the room from each other. On the next breath of air, the balloon grows even bigger and suddenly these people aren't even in the same city! Chaos ensues, the squiggly flat peoples' government breaks down; martial law is declared, and eventually the squiggly flat peoples' society reverts to despotism.
Now the squiggly people are dead or dying, hopelessly spread out across the vast area of their expanded, infinite universe. I hope you're satisfied!