The question that I have will it be possible if I use some kind of step up boost converter (not sure if it is the right name) to keep on using the battery and push 3v and above to the circuit ?.
It is possible to use a boost converter to increase voltage as required, though it calls for higher input currents which may exceed a battery's drain limit. In most cases it's better to use a series pack and buck convert down to the required voltage. Efficiency is better, input currents are lower, and there's a much wider selection of buck converters compared to boost converters.
As already stated you can't drive a Li-Ion cell much below 3V or it will cause damage to the cell. The operating range for a standard Li-Ion battery is typically 4.2 to 2.7V, but limits can vary somewhat depending on the particular make and model. For example the LiPo type cells used for powered models have a 3.0V lower limit. High drain 18650s can have a 2.5V lower limit. There's also high voltage Li-Ions that have a range 4.35 to 3.0V.
Li-Ion cells may or may not have protection built-in. When they have it, a small round protection PCB is encased under the shrink wrap typically on the negative side. The protection PCB limits voltage and current to safe levels. A protected battery will disconnect and drop out when limits are exceeded. When they don't have protection it's up to the circuit to ensure limits are not exceeded. Any Li-Ion battery must have protection either built-in to the cell or externally via the circuit it powers.