The bottom tranistor must conduct fully to have the follow on transistors (emitters) pulled to ground as far as possible.
Use an 820Ω base resistor for the bottom transistor for a full base current of 5mA. With this base resistor value you'll get the collector voltage down to 0.3755 with minor increase with the number of transistors connected.
With a base resistor of 10KΩ the collector voltage is 0.886V, just enough to have one LED bright.
Decreasing the values of the base resistors of the LED switching transistors will increase the collector voltage of the bottom transistor, thus reducing LED brightness.
Here is the schematic for three LEDs, all at full brightness of 19.5mA. Reducing the base resistance further will not decrease collector voltage, since the transistor is fully saturated already.
Boncuk