This purely my subjective point of view. Other people will have other preferences. There is no "right" answer to your question.
If you're writing for an English exam such as Cambridge FCE or CAE levels, please stop after third or thirdly. I've read fourth and fourthly in papers (and elsewhere), and that for me is teetering dangerously on "boring" and "mechanical".
To list a series of points or observations in an essay, a paper, or a formal letter you can start with either First or Firstly; then second or secondly and then use a different discourse marker such as :
- (to contrast) on the other hand; nevertheless; however;
- (to compare) similarly; likewise; in comparison
- (to add) furthermore; in addition; moreover.
- (to conclude) finally or lastly.