Tau = R * C => 100 us
Full charge/discharge occurs at, say, 6 Tau or 600 us
Max frequency to see full cycle => 1 / 0.0006 = 1.67 kHz.
As you use higher frequencies, you won't get full charge or discharge waveforms.
Hit the circuit with 1kHz and change the time/div on the scope to get just one charge and one discharge. The exponential nature will be more apparent when time is spread out.
Using 10k will change things, you would want a maximum frequency of just 167 Hz to allow enough time for full charge and discharge cycles.
There's nothing magic about 6 Tau, 4 or 5 Tau may be workable. It depends on what you want to see.
As to the last trace, I'm not sure what is going on there. What is the yellow trace? Is that the input?