We say "in a few years" but "in the past/last few years". Then how can we express " the period of 100 years leading up to now"? Which is correct, "the last/past one hundred years" or "the last/past hundred years"? Or both?
The two could be regarded as saying subtly different things, so the choice is yours. Neither is incorrect. However, a hundred is a thing for when there exists an English word, to which one may licitly refer as well as one may refer to a dozen. That is, a dozen is not merely a number in sequence between 11 and 13, but represents a poetically complete unit of a certain sort. The hundred is like that, as well.
Do you remember the Mudville Nine?
Therefore, on Orwell's principle to eliminate words one can do without, "the last/past hundred years" may be preferable, unless you feel that you have reason to write otherwise.
(Incidentally, if the period in question were 153 years, then it probably would not be ideal to write of "the past hundred fifty-three years." The usual editorial style in that case would be "the past 153 years" but, if one wished for some reason to spell the number out, then probably "the past one hundred fifty-three years." After all, it is unlikely that one would regard 153 as a poetical unit of completion like a dozen or a hundred.)
As @thb says in his answer, "The two could be regarded as saying subtly different things". I'll just elaborate on that.
If you wrote "the last/past hundred years", I would interpret that as probably meaning approximately the last hundred years, maybe going back as far as 1900, maybe even including the last 120 years.
If you wrote "the last/past one hundred years", I still would not interpret that as normally meaning exactly the last hundred years, but I might interpret it as less approximate (or more tightly intended) than the first version.
While I don't want to 'put words into @thb's mouth' that may be what he meant by "could be regarded as saying subtly different things."
If you want to make it clear that you are talking about a very approximate time period, you could write "the last/past hundred years or so". In this case, certainly omit the one.