The limitation is the fiber melting. It won't melt along the whole length - there will be some bend/crack/dirt which will make a hotspot, and that hotspot will then absorb more laser light and get hotter, and you'll get a runaway heating. The hot bit will then propagate slowly towards the laser source, eventually destroying the whole fiber.
High power fiber optic communication systems have protections in place to detect this happening and turn off to reduce the damage radius.
Laser cutting machines use optical fibers and can push tens of kilowatts if power, although I don't know what sort of fiber arrangement they do use. But I think that still indicates that you can push a lot of power through fiber.
I don't really have numbers for you, but there are currently 120 kW+ fiber lasers. A lot of the difficulty is in coupling, small defects or misalignment can easily self-destruct. (https://www.ipgphotonics.com/en/products/lasers/high-power-c...) Commonly used QBH connector (https://www.coherent.com/news/blog/qbh-fiber-optic-cables) The losses inside the fiber can be small in comparison and less of a problem due to manufacturing process control. There are efforts to develop hollow-core fibers that may be able to couple into free-space with lower loss and risk.
It's worth noting that the word "fibre" should be interpreted very loosely when it comes to high power lasers. They are often very specialised and in many cases more like "rods" i.e. held completely straight as even small bending would introduce enough loss to then cause the fibre to melt.
That said you can deliver 100s of W with relatively off the shelf fibres.
