There are two discs having different radii and having friction at rim. One of the discs is rotated and the rims of both discs are brought in contact . At last both discs will gain some angular velocity. In this case why can't we conserve angular momentum.

one above the other ?

Then we can apply the ppl,   as friction is an internal force and torque due to friction is also internal.

When the rotating disc is brought in contact with the stationary one for some time one disc will slide over the other and there will be a loss of energy. Later on both will start rotating in a resultant angular velocity. May be this energy loss is the reason for not conserving the momentum

angular momentum  surly conserves about a point , about which net external torque is zero so see point in the system oor out of it and apply conservation of angular momentum

NOPE.We cannot conserve angular momentum,because friction acts on the rim due to the ground also,which gives an external torque

First of all u have to define a system and depending on it I can tell u whether u can conserve angular momentum or not !! ( Then those 'external ' and 'internal' phrases will also becomee clear ) .

Suppose in my system I include both the disks . So what are the forces acting on THIS system ?

Friction clearly becomes an internal force so in the doubt we can eliminate it and the other is their wt s.

Now what is the torque of their wt about their common axis ?

On a first thought it will be clear that the torque is zero . So NO external Torque is acting on THIS system about the chosen axis . So We CAN Conserve angular momentum ogf THIS SYSTEM about their COMMON AXIS .

But if anybody wishes to include just a single disk as his system , clearly an external torque due to friction acts on his system and he can't conserve the angular momentum of his system about their common axis .

I hope it is clear now !!