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Hi,
Not sure, just read circuit has several limitations, one being 555 current limit of 200mA; another being timer Vout is always 1 - 2V below V+ (3.7V * 2) - ~2V... - and the load influences that; also, charge diode-capacitor pumps are weak current-wise. I breadboarded a voltage inverter and a doubler with a timer and diode-caps months ago out of curiosity and I doubt I would use either in a real application, especially one that required more than a few mA, but each to their own preferences. Give it a go in a simulator or breadboard it and find out if it's suitable for the circuit needs. Have you seen the version using a complementary pair on the timer output? There's also a version that parallels two timers outputs to "double" the voltage.
A cell phone has a charger circuit in it that has a 5VDC USB input and charges the "3.7V" lithium battery cell to 4.20V. A voltage doubler is not used.
A 555 was used as a voltage doubler about 30 or 40 years ago. The voltage was not doubled, instead it was increased only a little.
A voltage boost converter circuit using a Mosfet and an inductor is used today, but not as a battery charger, it can be used to boost the 3.7V from a lithium battery cell in a power bank to the 5VDC USB used to feed the charger circuit in a cell phone.