What would be the results of two bridge rectifiers in series?
The bridge rectifier can be thought of mathematically as taking the absolute value of the input voltage.

Taking the absolute value again does... nothing. The only difference of adding a second bridge is you are out 4 more diodes and the output voltage is 2 diode forward voltage drops lower.

Pros


Make diode manufacturers happy by buying more diodes
Keeps your circuit warm by dissipating heat in the second bridge
Slightly lower output voltage (by 2 diode forward voltage drops, though there are simpler ways to get this)


Cons


More parts in your design (more board space, money, assembly time...)
Consumes more electrical power, lower efficiency, increased heat dissipation requirements...
Slightly lower output voltage (by 2 diode forward voltage drops, though there are simpler ways to get this)


Joking aside, I can think of no practical reason to do this. If you want a smoother DC output, add more/larger smoothing output capacitors, not a second bridge.