When the output V is the same as supply V, it suggests the chip is no longer oscillating. It may stay on, which drains the supply. Or it may stay off, which lets supply V through unhindered to the output stage.
It is a big jump to boost 1V to 3.3V. Whatever the load requires, a boost converter must draw 4 or 5 or 6 times that much current from the supply. Can your 1V supply provide that much current? Can you confirm that it doesn't droop?
Things to try:
Try attaching a higher resistance load. If the circuit continues to work, then increase the the load in steps to see when the circuit starts to misbehave.
Try reducing the value of the smoothing cap. This will produce more ripple on the output, so the chip may detect the variations more easily, and respond.
Try increasing the coil Henry value. This should reduce frequency, possibly making it easier to sustain oscillations.
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For curiosity sake I made a simulation using your values. I adjusted it to provide 3.3 V to a 180 ohm load.
This screenshot shows the waveforms you can expect (theoretically).
With your 22 uH coil, a frequency of 300 kHz is reasonable.
I installed small value resistors which represent a certain amount of resistance in the supply and switching device.
The peak drain from a 1V supply is 123 mA. The average is about half that, but it still amounts to several times what your load draws at 3.3V.
One other thing. On startup it is possible for a control IC to get locked at either a 0% or 100% duty cycle, in the attempt to reach your desired output volt level. This can prevent proper operation because the switch stays On or Off with no switching action. I don't know what your NCP1400 is designed to do in this regard.