Infinity is many things to many people. To mathematicians, it is something that is practically 'concrete' and the many different kinds of 'infinity' can be classified in terms or Cantor's Transfinite Numbers. There are countable and un-countable infinities, and who knows what other kinds in between.
To physicists and astronomers, there are no infinities in the physical world. Not even space and time itself need to be 'infinite' in the purely mathematical sense. Whenever 'infinity' appears as a prediction by a theory of the physical world, it is deemed a flaw, and must be eliminated. It is considered a signal that the particular theory has been extended beyond its domain of applicability. Newtonian mechanics had no problem with bodies attaining infinite velocity, or acting through space with infinite speed. These expectations led to falsifiable predictions and that is, in part, why we now have special and general relativity. In the few cases where general relativity predicts 'infinity' in the form of singularities, this is regarded as a defect in general relativity theory that will be healed when a fully quantum theory of gravity is developed. For more on singularities, see a Previous Question number 1503.