The phrase "slightest degree" means the smallest amount or level possible. It refers to the tiniest or most insignificant extent of something. Full definition
Even slight degrees of dehydration can have a significant influence on your metabolic rate and thus how many calories you burn off on a day to day basis. (sixpackfactory.com)
I've only ever seen this before on some Figoni et Falaschi teardrop coupes and to a much slighter degree. (automobilemag.com)
Although it would be infeasible to create individual - crafted emails for each of your subscribers, there's no reason not to identify clear sub-niches within your subscriber base, which could benefit from even the slightest degree of personalization. (smallbusinessesdoitbetter.com)