Today I had the idea of creating a very simple discrete battery charger based on
this circuit bigclive showed in his video since it already had that really cleaver "cell in" indicator trick, but I wanted to add a "charged indicator", so I just added another transistor to indicate just that and light a green LED:

I've done some simulations and also breadboarded the circuit. It works quite well, but as you might have already guessed the LED starts to turn on before the battery is completely charged, but that's fine all you need to do is check if the LED is bright and not dimly lit.
Another idea that I had was to add a another transistor to switch off the charger when the battery is full, but I still haven't tried it out yet:

I just thought it was a nice little circuit to share if anyone is interested. Feel free to let me know if you liked it, suggest any improvements, or point out any mistakes I've made.