The behavior you are describing is essentially bi-stable, meaning you want to transition between two states. For that, you need some way of storing & acting on a state. If you're simply wanting to avoid IC's, you can use a switch (or relay) to hold this state change. Using a relay, you can select a dual-throw type and allow the state of the relay to dictate charging & discharging. By doing the charging/discharging through the relay's activation coil, you will get state change once the current drops. Changing the resistance between the charge & discharge paths allows different slopes for your waveform.
Relays wear out after lots of toggles, so a better way would be to use at least two transistors (like BJT's) and again setting the 'toggle' for the state change by having one discharge and the other charge. You can do this by having the base of the transistor biased off a voltage divider tied to the capacitor. Of course, if you're going to go much beyond two transistors then you're really beginning to push what makes sense for avoiding an IC.