The max swing is roughly 10- (FLOOR (LOG(10) Popultaion)) * 5
Most populations range between the single billions (log = 9) and the single thousands (log = 3). So for the highest population systems, the faction swing can be about (10-9) * 5 = 5% a day. For the lowest population systems it can be (10-3) * 5 = 35% per day. The function to reach the cap is also logarithmic or asymptotic, meaning that the amount of work you need to do to get a larger swing grows the closer you get to the max. So you might be able to get 50% of the max with only a little effort, but getting 99% of the max would take a much MUCH larger amount of work.
Some activities are not capped, AFAIK. Whether by design or by bug, activities which reduce influence, such as murder, smuggling and trading at a loss have not been capped in the past. A few patches ago, the negative influence for trading at a loss was removed, but undesired bizzaro BGS effects have come back from time to time between minor patches. I can't say authoritatively what activities are still uncapped, but I think at the moment it is only murder and smuggling.