While we wait for Google to educate us more on how crawl budget works at Google, more and more questions and answers are popping up around the topic.
Google's Gary Illyes and an SEO named Idan Ben Or have gone back and forth on Twitter about using the nofollow attribute to help with crawl budget issues. Gary Illyes basically said it depends on how large the site is.
For a site of a few thousand pages, Gary said no - it won't help much. But for a site of a 100,000+ pages, it may help but it depends.
Here are those tweets:
Forum discussion at Twitter.