How do I stop/compensate for a voltage drop across a transistor?
Where are you measuring this amplitude drop?

If you connected the 555 output directly to the base of the transistor, and the transistor's emitter is grounded, this drop is expected, and you are stressing the base-emitter junction of the transistor.

The base-emitter junction is effectively a diode and will hold the base voltage to about 0.7 volts above the emitter (depending on base current).  To avoid damage to the transistor, you need a resistor between the 555 output and the transistor base to limit current through the base-emitter junction.