In terms of the portraits you referenced, Louis XIV would have sat for an hour or two a few times throughout the process, starting with sketching sessions with the artist, and progressing to the full-scale painting. That said, the vast majority of portrait painting was done without the sitter (monarch or otherwise) present.

Once a suitable likeness had been established to the sitter’s satisfaction, the sketches would be translated to the canvas and the artist would be doing most of the work in his or her studio. Clothes may have been loaned to the artist to put on a mannequin or to have an assistant/model wear while posing, or the artist would have their own collection of clothes to reference. But typically, the artist worked the majority of the painting without the subject there.

Another common practice for monarchs was distributing a face template to artists. These were guidelines set by the monarch for how their likeness was to be represented in portraits. Queen Elizabeth I made use of these templates throughout her reign, most typically using stylized features based on her face when she was in her 20s, requiring them to be used well into her 60s. This wasn’t unique to her, however, nor was it as simple as vanity. (One gorgeous example of an artist using a face template is this portrait of QEI currently in the collection of the Weiss Gallery in London.) The likeness of the monarch was an official document, and as such, it had to be represented consistently, which was trickier back before the days of photography (lots of variation in skill from artist to artist, as you might imagine. For every great master like Holbein there were dozens more very average painters who nonetheless were in the employ of the court at any given time). These face templates gave artists who received commissions for official portraits, but may never have had the ability to sit one on one with the monarch, a way to consistently depict their likeness.

In 1563, just over five years into Elizabeth's reign, and presumably after the present portrait type had been disseminated, Sir William Cecil drafted a proclamation designed to control the production of the monarch's image, forbidding further portraits of Elizabeth being made until an appropriate model (in the form of a face pattern) could be provided to artists to copy from. After this, ‘hir Majestie will be content that all other painters, or engravers…shall and maye at ther pleasures follow the sayd patron or first portraictur'. — The Weiss Gallery at London Art Week

These face patterns were used well into the 18th-century for heads of state, so it’s quite possible that Louis XIV’s portraitists made use of them as well. In the case of Hyacinthe Rigaud, the painter responsible for the portrait of Louis XIV you reference, it is equally probable that he would have been granted sitting sessions with the King, as he was a very big deal as an artist and one of Louis’ favorite portraitists. But either way, Louis wouldn’t have sat for the entire painting process.