I measured the base current with a multimeter, I measured 430 milliamps base current. And the tip110 transistor will still spin the motor without the arm or any load.Calculate the base current:
1) Base voltage for a TiP110 is typically 1.8V on its datasheet.
2) The 4.2V battery is almost dead at 3.8V.
3) You have a 1 ohm resistor limiting the base current.
4) Then the base current is (3.8V - 1.8V)/1 ohm= 2A.
5) If the battery is working then you destroyed the base of the TIP110 because its maximum allowed base current is only 50mA.
For a load of 2A then the base current is shown to be usually 2000mA/250= 8mA, then the base resistor is (4.2V - 1.8V)/8mA= 300 ohms.
The saturation voltage loss is 1V to 2.5V so the motor gets 3.2V to 1.7V when the battery is fully charged at 4.2V.