In the classical inflation and big bang theory how can the universe grow to infinity? Some astronomers make the conjecture that the universe is actually infinite in extent (which is a bad manner of expressing, I know). If so, how could it grow to this size in the classical theory of the Big Bang followed by cosmological inflation?
 A: Infinity is a mathematical concept, as well as the concept of variables describing dimensions. 
Physics is about observations, either in the laboratory or of the cosmos, which are fitted with mathematical models. It started with the geocentric system, became  the heliocentric system and then the realization that the galaxy is composed out of sun like stars, and that not only galaxies  but also clusters of galaxies exist in the cosmos, lead to models with newtonian gravity fits. These failed when Hubble's expansion was observed and General Relativity was validated at the observational level .

In the classical inflation and big bang theory how can the universe grow to infinity?

What we have is the observable universe, which is the interpretation of the available observations and data of the universe:

The observable universe consists of the galaxies and other matter that can, in principle, be observed from Earth at the present time because light and other signals from these objects have had time to reach Earth since the beginning of the cosmological expansion.

The Big Bang model, is a GR mathematical model which tried to explain the Hubble expansion with an original singularity about 14 billion years ago , so as to be consistent with the data of the observable universe.
The model keeps the name of Big Bang, although it has further morphed, using quantum mechanics for the times around the beginning of the universe, where the singularity was posited in the original BB.
So the "how" the variables can go to "infinity" is a mathematical projection that can be examined on the current model.

If so, how could it grow to this size in the classical theory of the Big Bang followed by cosmological inflation?

If you read the link you will see that the present model is not really a classical model, but a hybrid of an effective gravitational theory and classical General Relativity. Making the variables of models go to very large numbers, including mathematical infinity is a simple  mathematical game. Depending on the model parameters, the answers will be different.

The ultimate fate of the universe is a topic in physical cosmology. Many possible fates are predicted by rival scientific hypotheses, including futures of both finite and infinite duration.

It is the mathematics ........
A: That is not quite that simple. There is theory and measurements, and it places some constraints. There is more. 
First, if the universe is infinite now it was infinite at the Big Bang. You can have an infinite universe and have it all in the spacetime at the Big Bang. It does not grow to be infinite, it either is or is not. (Ignoring multiple dimensional string theory). The scale expands after that, but the total size if infinite would still be infinite. A better term used in the literature is 'unbounded'. Being unbounded or not is not dependent on the expansion. The Big Bang occurred everywhere at once. It is not that it will expand to infinity, or in an unbounded way, if it will be it is already unbounded. (Unless the topology could change, that is unknown. More on topology below).  
Second, according to GR and the standard (Lambda-CDM) cosmological model with parameters pretty well measured in Planck 2015, the universe is very close to flat. The cosmological curvature was estimated to be very close to 0, with the density parameter predicting it being 1 within 0.5%. If it was exactly 1 the universe is exactly flat. It could still be open or closed instead, by very very little, because of that uncertainty. The estimate could get better, but if flat it still could any of the 3 possibilities within the uncertainty. 
If the universe was closed it would not be infinite, or unbounded, the simplest topology would best spherical, and closed, but since so close to flat, it could be humongous. 
Of course our observable universe is about 93 billion light years (where the matter that emitted the light we see now, is now, in our comoving frame). We are talking about all the other light and matter which emitted it that has not had time to reach us since the Big Bang.
If it is flat, exactly, or if it is open (i.e., if normalized curvature is K=0 or 1), whether it is unbounded or not depends on the topology of the universe. The trivial topology is E3 and would have it be unbounded, for flat space. If negative curvature could be instead a negative curvature hyperbolic topology. If it is strangely connected, with boxlike topology like PacMan, light and matter could go off one side and enter into the other. Or it could be not simply connected in other ways. It could have a 3d torus topology, it'd be flat but finite. Again, the simplest topology for flat is E3, unbounded. 
Either way, which whoever topology, the universe expansion will continue forever if it is open or flat. If closed it'll recompress.  
It is true we may never know, but there is some observational data possible for the PacMan universe, eg, light from one Galaxy coming one way and the other way. Not a single hint of that has been observed, but of course it could be the box is much bigger than the Hubble size, and we have not see it yet. Still, at least it looks like our observable universe and somewhat beyond it has, so far, an E3 topology. You can bet that if somebody discovers the same Galaxy on one side and the other, and then it is confirmed with others, yes they will get a Nobel. Whatever the reason is.
