In flight: Not a thing. As far as the a/c is concerned there isn't a crosswind - it all arrives from the front. Presuming no sideslip, of course.
On the ground: No great drama as long as the control surfaces are held appropriately and depending on just how much over the limit (1kt? 10kts? 100kts?). There will be a tendency for the a/c to weathercock into the wind regardless. At some point taxiing becomes impossible to the strength of wind overcoming steering authority &/or ability to keep the a/c from becoming airborne or getting flipped. That applies no matter where the wind is from.
Take-off: Ditto no great drama depending on just how strong the x/wind component is. Hold the ailerons to prevent the upwind wind from lifting, rotate briskly to minimise the transition period from tyre borne to wing borne so that the tyres don't get dragged sideways as much. The a/c will tend to yaw into wind as soon as airborne. As rough rule of thumb whatever it yaws to will be approx. the drift allowance needed to maintain RWY track if your climb speed is similar to the airborne speed.
Landing: Becomes more of a problem, again depending on just how much over the demo. limit the x/wind component is. As others have said it's rudder input that becomes the limiting factor.
Using a wing-down/sideslip method then as speed reduces more rudder input will be required to maintain HDG. This is compounded because as speed reduces the drift increases so even more wing down will be needed to maintain position over the centreline. That in itself requires additional rudder input. Eventually you'll run out of rudder & the a/c will start to turn in the direction of the low wing.
Keeping position over the c/line isn't really a problem (big wing compared to rudder so it can generate a HUGE amount of sideways force to counter the x/wind). Use more wing down as speed reduces/xwind component increases.
Once the wheels are on the ground then the addition of nosewheel steering + differential braking assists control.
You can partially reduce the problem by limiting flap amount. The extra speed throughout the flare/hold off/touchdown reduces the drift for a given wind strength & makes the controls more effective.
Another option is to use the crabbed technique: A moment before the wheels are about to touchdown on the runway apply sufficient rudder to yaw the a/c (a flat turn so opposite aileron will be needed) so that the a/c points down the runway just as the wheels contact the surface. The disadvantage is that there's little margin to correct a misjudgement, unlike the wing down method.
The problem is that the rudder can only apply enough force to yaw the a/c at a certain maximum rate. The rate reduces as speed reduces due to reducing rudder effectiveness. The stronger the xwind then the greater the crab angle needed to track along the runway. That means more time is needed to yaw the a/c.
As soon as the yaw is started then the a/c will begin to drift. At some point the time needed to yaw to align the a/c vs. the amount of drift that develops throughout the process will limit how much xwind is feasable. If you like, you can think of it as running out of rudder too ie not enough rudder to yaw the a/c fast enough.
I tend to combine methods. I leave the yaw until shortly before wheels touch and also apply wing down. Sometimes I've also had to differentially close the throttles to increase the yaw rate.