How can a Ultralow Voltage Step-Up converters start up?
I studied the patent that was briefly briefly shown to us by user @IgnacioVazquez-Abrams to the extent that I begin to understand the operation of it. The patent doesn't exactly describe the operation of LTC3108 but something very similar to it. Here's what I made of it. Feel free to offer any corrections.

Consider the following picture, taken from the patent:



At the heart of it, there is Q1, a junction FET whose threshold voltage is very near zero volts. So near, in fact, that ambient noise of the current source (thermopile or whatever) causes changes in the conductivity of Q1. This causes changes in the current through the primary winding of the transformer T1, which transforms the changes in current to changes in voltage at the secondary winding, at the ratio of the windings of the transformer, for example 1:100. These are then fed back to the gate of the FET, in the case of this schematic at an inverted phase, so that Q1 will further amplify the noise oscillations. The net effect is that whenever sufficient current is provided by the current source, the circuit formed by Q1, C1 and T1 quickly begins to oscillate at a roughly known rate.

As the oscillation starts, an AC voltage is formed at the secondary winding of T1. This will then be rectified, regulated and used to power up the circuitry.