What is the ruling on joining two prayer beads (masbahatayn) together to increase the number of beads to 150, in order to keep track of the number of times one says tasbeeh, istighfar, and hamd, due to forgetfulness and old age?

There is no harm in joining one rosary (misbaha) with another to increase and control the count, and to use them both for glorifying Allah (tasbih). There is no specific number for the beads of a rosary. It has been authentically reported that Abu Hurairah had a string with a thousand knots on which he performed tasbih. The more preponderant opinion is that performing tasbih with the hand is superior to using arosary.