If you can only use opamps and R's and C's:
You could use a switch and capacitor is series, with the switch connected to the + rail, and the capacitor connected to ground.
An op-amp could be connected as a comparator to see the voltage on the capacitor.
Thus, when the switch is pressed, the capacitor will be rapidly charged to the rail voltage (know as a sample and hold). (in reality
stick in a few ohms resistance to prevent damaging the switch).
The high input impedance of the comparator means that it won't discharge the capacitor in a long while, so the comparator
output will be high for ages.
Use the comparator output to slowly charge another capacitor, this time through a resistor. Use a second comparator connected to
this second capacitor, to detect when the voltage reaches a threshold (which can occur 10 seconds later).
Use a specific value resistor across the first capacitor, to slowly discharge it, so that it is ready for another switch press later.