Freelancers, do you bill for time spent filling knowledge gaps?

Generally speaking I account for it, so yes. If someone comes with a problem they've not been able or need to solve and I basically know what tools I can use that can solve it, but I've not spent the time to learn how to use them for this particular problem, yeah, they're going to have to pay for that.

For example, they want a chart on their web site using chart.js but don't know how to get and parse their data and know I can do that with Perl and some regex work. I'm not a pro at that but I know I can generally find out what I need to know to make it work for the task in hand.

I will certainly never use that regex again. They will probably use it for a very long time, or save a ton of time using it once.

So, yes, they would have to pay me to learn what I need to know to complete that task.

I would say learning and closing gaps is always part of development and must be payed. Let's say you are the most competent C++ developer in the world and have the whole STD memerized (no real human is this guy I think), but part of the current project is using any other library or framework out there, then you need to dive into that documentation. And epecially as freelancer you need to learn how the companies code base every time you switch the customer or project.

Another thing is if you want to qualify for certain freelance jobs. Let's say Qt is a thing you want to do professionally. Then it is very likely that you have to build up some knowledge and portfolio to get such a job. Then the freelancer pays for the time obviously.

I do if I need to spend time learning something specific to their application or business. And I tell the customer that up-front. I may propose alternatives that I am more familiar with. A couple of times the customer has assigned one of their own developers rather than pay me to learn something, but that’s been rare.

If I can resell it, or services made possible or higher value by having made the investment, generally no.