The question you are asking had a consensus answer that agrees with yours until the end of 19th century. All infinity is like "continuous generation of numbers", or what philosophers called potential infinity, there is neither physical actual infinity that could result from completing such a process, nor even mathematical one. This view was expressed most definitively by Aristotle to resolve Zeno paradoxes, and other difficult issues, and was almost universally accepted for centuries. There was even a coined Latin expression for it, "infinitum actu non datur", the actual infinity is not given.
There were dissenters, notably Leibniz admitted actual infinity of objects, but not collectible into a set, see Are infinitesimals in the Newton and Leibniz calculus potential or actual? Bolzano allowed the collecting but not comparison of infinities using bijection. Both were motivated by upholding the Euclid's part-whole axiom, "the whole is greater than the part", see Mancosu's historical survey Measuring the Size of Infinite Collections for other instances, but they were few and far between.
The consensus was challenged head on by the founder of modern set theory, Georg Cantor, at the end of 19th century, and eventually rejected, at least in mathematics. And Cantor did not spare even the part-whole axiom, Dauben writes:
"A typical argument used by Aristotle and by the scholastics involved the "annihilation of number". Were the infinite admitted, it was said that finite numbers would be swallowed up by any infinite number or magnitude. For example, given any two finite numbers a and b, both greater than zero, their sum a + b > a, a + b > b. However, if b were infinite, no matter what finite value a might assume, a + b = b, and this seemed contrary to a well-known and basic property which the addition of any two positive numbers ought to exhibit... Infinite numbers were consequently rejected as being inconsistent...
Cantor condemned this kind of argument, however, on the grounds that it was fallacious to assume that infinite numbers must exhibit the same arithmetic characteristics as did finite numbers. Moreover, by direct appeal to his theory of transfinite ordinal numbers, Cantor could demonstrate that infinite numbers were susceptible of modification by finite numbers. In fact, the distinction between Cantor's w and w + 1 showed expressly that finite numbers could be added to infinite numbers without being "annihilated."
Taken literally, Cantor's transfinite numbers suggest that theoretically a "computer" could keep counting for the actual infinity of time, and then some, "to infinity and beyond". As for physical existence of actual infinity, of space, of time, or of physical objects, the question is much trickier. They are no longer considered impossible, but it is hard to say what it would mean empirically that space is "actually infinite". There is no "test" that can tell us if it is so. But finite space does not need to have an edge, imagine the surface of a sphere, it is finite, but has no boundaries. And it can exist on its own, without sitting inside a three dimensional space. It is assumed that our universe is something like that in three dimensions, so we can not walk up to the "edge" and see it expand (and there may be no external anything for it to expand into). Similarly, under some theories of Big Bang the universe existed eternally, i.e. for an actual infinity of time, and under others it only existed for finite time. But again, there is no empirical test that can tell us one way or the other. So it is not clear if "physical actual infinity" is even meaningful.