37 Wikipedia lists large scale numbers here. As only the 10 x with x being a multiple of 3 get their own names, you read 100,000,000,000,000,000,000 as 100 * 10 18, so this is 100 quintillion in American and British English and 100 trillion in most (non-English speaking) other places.
The type of writing you are doing also plays into your decision. For example, in legally binding documents, like contracts or exhibits to contracts, the spelled out number is the legally binding number. So if a text said that, "you are 99% (one-hundred percent) responsible", the 100% number would be legally binding, not 99%.
Numbers with more than 100 zeros - English Language Learners Stack Exchange
And the usage always seems to involve a number between 100 and 200: "a buck fifty" and so forth (the term seems to be wedded to the indefinite article: "a buck something ").
The flow rate increases 100-fold (one hundred-fold) Would be a more idiomatic way of saying this, however, the questioner asks specifically about the original phrasing. The above Ngram search would suggest that a one hundred has always been less frequently used in written language and as such should probably be avoided. Your other suggestion of by one hundred times is definitely better than a ...
As you note, "percent" means "for every hundred," so there is nothing at all wrong with percent values greater than 100 when discussing proportions, e.g. profits increased by 120%. Like fractions, however, percent values are longer to express than multiples, so for values much greater than 100 you may hear them less frequently: