Is there a logical fallacy here? "The universe could be infinite in size".

Joe's  position is contradictory. ambiguity fallacy A finite set is measurable by definition. If immeasurable,  it is infinite. If the set is increasing in size faster than one can count doesn't mean it is infinte; just count faster. (Just because the hare couldn't catch the tortoise doesn't mean the tortoise was running infinitely fast.)

This question involves an incomplete understanding of infinity. Infinities come in two shapes and sizes : Uncountable and Countable. In fact, countability and infiniteness are different concepts.

So the universe would be a countable infinity in the sense that everything can be counted but infinite in the sense of natural numbers

Too many undefined terms here to begin to address the logical form.

It doesn't seem to be a fallcious argument, but it's not coherent.

Even if it was possible that new objects could be made faster than could be counted, due to the fact that there is a set amount of energy and matter in our universe (refer to the law of conservation of energy) for every object being "created" means energy or other objects are being converted into those new things at that same rate. So you wouldn't end up with any more mass and energy than what you started with. 

 Unless you throw the law of conservation of energy out the window, the example fails to prove the universe is infinite.