What is the pressure (in bar) in the fuselage at flight ceiling of large passenger airliners?
Is this pressure the reason for the difficulty in having large windows? If so, how can the window size be increased?
The pressure inside the cabin is usually more than the outside pressure at the cruise altitude. The pressure altitude (ISA altitude with the same pressure) inside the cabin of civil aircraft is called the equivalent effective cabin altitude or normally cabin altitude.
As the aircraft climbs, the cabin altitude is gradually raised so that it reaches around 8000 ft for the aircraft cruising altitude of ~40000 ft. This level is maintained at flight ceiling too.
The pressure is around 0.75 bar inside the cabin. The outside pressure at this point is around 0.2 bar, resulting in a pressure differential of 0.56 bar. This pressure differential causes stresses on the aircraft body.
In addition to the stresses, pressurization - de-pressurization cycle can cause fatigue in the windows.
As reducing the cabin altitude will improve the passenger comfort, newer aircraft tend to have lower cabin altitudes. For example, the Boeing 787 has a pressure altitude of around 6000 ft and the Bombardier Global Express jet has a cabin altitude of around 4500 ft.
This is achieved by the use of advanced composite materials, which allow a higher pressure differential and have excellent fatigue properties compared to the metals. As a result, the windows of newer aircraft are considerably larger than their predecessors.
What is the pressure (bar) in the fuselage at flight ceiling of large passenger airliners?
Aircraft are typically pressurised to an equivalent altitude of 1800m (0.815 bar) to 2400m (0.750 bar). Be aware this is absolute and not relative to the atmosphere around the plane.
Is this pressure the reason for the difficulty of having large windows?
It is the most important reason. When stuff is not pressurised you can pull off fairly decent stunts, like the huge mirror on the SOFIA aircraft for instance.
What advancement would be a game changer and allow large windows?
Composite materials are currently the big deal and bringing considerable benefits (See this question). If you can skip the material change entirely, by using a thing such as transparent composites, then you can make them any size you please.