Circuit-wise, the NPN's T1 thru T6 are emitter-followers. In order to turn 
them on, the base-emitter junction needs to be forward-biased, which is 
about +0.7 volts. So, if the base is driven to +5v (the max possible from 
the 74HC device), the max-possible voltage at the emitter will be 5.0-0.7 = 
4.3 volts. It makes no difference how high the collector voltage is (unless 
it's SO high that you get breakdown).

You are correct that the 74HC will source base current, but that will only 
happen as long as the base-emitter junction is forward-biased. For the sake 
of argument, let's imagine the base is at +5V and the emitter is around 
+160V. In this case, the base-emitter junction is reverse-biased, which 
means the transistor is off and no collector current (other than leakage) 
is possible. Stated another way, it's not possible for the circuit as drawn 
to produce more than +4.3V at the emitter.

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