Generate a higher-than-power-supply voltage with capacitors
Your idea has been invented before, and it works: the circuit is called a charge pump. You can find many charge pump designs on the Internet. They can be simpler than yours (see below).
As for your immediate question, why the voltage on C9 doesn't go higher than 4V, even though the top end of C7 goes all the way up to 8V:
Transistors are not just switches where the switch goes on if the base has voltage - they get way more complicated than that. In particular, note that the transistor detects the base voltage compared to the emitter, not compared to 0V. If the base is 5V and the emitter is 4V, the transistor only detects a base voltage of 1V, not 5V.
And that's exactly what's happening here. Because Q2's base voltage can't be higher than 4.54V (10/11 of 5.0V), Q2's emitter voltage can't go any higher than about 3.9V before Q2 turns off. Transistors are analog devices, so Q2 will find an equilibrium, where the base-emitter voltage is exactly enough to stop the emitter voltage going any higher.
This is used in &quot;emitter follower&quot; or &quot;common-collector&quot; amplifier circuits: the emitter voltage tracks the base voltage, minus about 0.6V. But that's not what you want here.
Note that Q12 doesn't have this problem, because its emitter is always 0V. For good switching, you want all of your transistors to be like Q12. If you want to have a switch connected to +5V, you can also use a PNP transistor, which uses opposite voltages, and will turn on when the base voltage goes below about -0.6V compared to the emitter, which you connect to +5V.
Q11 is used in &quot;reverse active&quot; mode. When it's turned on, current flows from the emitter to the collector, which is backwards. Transistors can operate in this mode, but not as well as they do in the usual &quot;forward active&quot; mode.
Note the whole C8/Q13/Q11/Q12/etc circuit is unnecessary. You can just connect the bottom of C7 to the output of the inverter, unless your inverter is too weak to supply the load current - in this configuration all of the load current comes out of the inverter. Q10 doesn't need active switching and can just be a diode. Q2 also doesn't need active switching and can just be a diode. If you make these changes, you will find yourself with something like the standard charge pump voltage doubler circuit:



simulate this circuit &ndash; Schematic created using CircuitLab
BTW: Different charge pump designs can also be used for voltage tripling, etc, and for creating negative voltages.