Characterization of RF Amplifier Chain
Well, that is an unsolvable problem, unless you either


use active components that have the same gain, but higher saturation points, or
dynamically change the gain (or dynamically add attenuation) based on your situation.


The second point is usually implemented by what is called AGC (automatic gain control), and will require you to design a control loop that is adjusted to the characteristics of your signal.

Also note that as a communications engineer, this problem is something you'll meet fairly often. A common solution is not to solve it: If the distortion caused by nonlinearity for very strong signals just makes them as good as "mediocre" signals, well, so be it, if you can still work with that. If you're in control of the power of the transmitter (and this isn't e.g. a radar system, where you effectively aren't), power control is a common thing to do (because more isn't always better).

Another option is to ignore the weak signals for now, and e.g. have multiple receivers, which, combining their receive signals, can achieve a processing gain sufficient to raise the weak signals "out of the noise floor". 

But all these methods are really higher-level things that require understanding of the signals you're working with, so we can't help you with that at this point – the first thing to do when resolving receiver sensitivity/dynamic range issues is to write down a somewhat comprehensive system requirement.