Chipped chainstay repair

Ouch!  I would say that both SingleSpeed and Yorkshire wallet are right.

Nail varnish works great.  You can generally get a good colour match and it forms a tough protective shell.  It definitely prevents any carbon threads from fraying, just take it easy and apply it in thin layers. Heli tape or a frame protector patch would add peace of mind.

Personally I'd do both of those and add a chain catcher for good measure (I know that's not the damaged part, but if you have shifting problems you can avoid the possibility of garotting your bottom bracket for the sake of a tenner).

Where the damage is seems a bit odd.  Is your chain too long or is the spring on the rear mech too weak or should you be using a longer cage?  I'm suspecting that you are on a compact chainset and that the chain has enough slack in it to get  "sucked in" to the chainring and stay.  If you can lose a couple of links, I'd give that a try, but depending on what your setup is, you may need a longer cage on the rear mech or a newer version with greater capacity.

After it's fixed stick some clear helicopter tape over the contact area.

I would learn how to set up your gears...or take it to someone who can.

Then suggest some white nail varnish.