A question about beefing up your mast really befits a different thread from one about mould on a sail but here goes anyway:
The standard mast does flex quite a bit and, while I have never personally broken a mast, it is not hard to believe that it might well break if used in very strong winds.
1. it is not likely to be used in these conditions because it just gets too uncomfortable and the risk of a gust ditching you into the water increases so most people (i.e. me) will prefer not to sail in too high a wind
2. when a sailboat leans to the wind the sail spills wind thus reducing the stresses on the mast - clearly a sailboat equipped with outriggers has less of a tendency to lean to the wind though it can still do so in which case the outriggers will submerge as the lean increases.
3. as the mast deforms the sail shape also deforms and should spill wind thus reducing pressure on the mast.
So if you are going out in strong winds, using your weight as balance to counteract the tendency of the boat to lean in strong winds and especially if your boat is equipped with outriggers you can expect to see quite a lot of bend in your mast.
IMO putting a bit of wood up the middle of the mast tube is not likely to do a lot in terms of mast rigidity - engineering principles state that the rigidity of a tube will be much the same as the rigidity of a solid rod because most of the rigidity derives from the outer part of the rod, not the centre so beefing up the centre will not have a whole lot of effect. But there are four possible solutions to this that I KNOW will deliver less mast bend:
a) don't go out in winds that are too strong
b) don't use outriggers
c) devise a roller reefing system so that some of the sail can be rolled away in stronger wind conditions
d) stay the mast to prevent it from bending
I use a,b and c on a regular basis. As to d) some time ago I devised a system of stays for the mast which are very effective indeed at preventing mast bend, maintaining sail shape (and efficiency) and allowing you to add more sail for lighter winds/greater speed in the right conditions. My stays are made of very light spectra line and were a very cheap addition that allowed me to use a jib as well as the mainsail. As I say, they really worked wonders for sail shape & efficiency but they add a lot of complexity to the rig and you need to be very methodical when using them. I haven't used them for a couple of years.
Hope these ideas help !