The limits of science depend how you define science.
Does it mean the study of physical things? Does it mean the study of things that are part of nature that may not be strictly physical? Such as numbers, or some views of consciousness. Does it mean acquiring knowledge through strictly controlled and testable hypotheses, even perhaps of non-physical things like ghosts?
In all cases, the answer is no, just in different ways.
If science is strictly the study of physical things, then it can't possibly comment on any non-physical things. It can't even comment whether non-physical things exist. Science has to remain agnostic on things like souls or gods. The farthest it could ever go is to say "so far as we know, the physical universe is a closed system". And if that statement ever proved to be false, it's debatable if science as a discipline would even be capable of acknowledging it. Science might carry on as if it never happened. Science as the study of physical things has to assume that only physical things exist, or it doesn't work anymore.
Science sometimes tries to pull other things under the umbrella of "nature" that aren't exactly physical, like numbers or certain concepts of consciousness. But I don't know what that would mean exactly. Seems to me that a thing is either physical, or it isn't. If a physicalist model can't account for everything, then it's either limited, or wrong.
Finally would be the version of the scientific method that openly claims non-physical things are open to scientific testing through hypothesis and experiment. You see this with claims that religion is a failed hypothesis, or trying to prove certain forms of morality through science, or disproving ghosts. I'm not sure what to make of this. Seems to me like whatever territory you're in at that point, it isn't exactly science in that sense. And anyway, there are plenty of "scientific" beliefs that aren't directly observable. Lots of things in astronomy, for example, are inferences based on other things. Like the effects of gravity and theoretical models. They can't exactly be isolated in a lab. Are those science? Well...
In any case science requires certain assumptions. It requires you to trust in human reasoning. It requires you to trust your senses. It requires you to believe that the universe is orderly and intelligible. Those aren't unreasonable assumptions, but science can't bootstrap itself into proving them.
Thank you for coming to my TED Talk.