It is fair to say that from the mid-late 1800s, there was an obsession amongst gunsmiths concerning revolvers. This obsession was increasing the number of shots a revolver can fire. Revolvers could usually only fire about 5 or 6 shots before the weapon had to be reloaded - this was admittedly preferable to the old flintlock and percussion muzzle-loaders of the early 19th century, which were all single-shot. However, gunsmiths all over the world still took it upon themselves to design revolvers that could fire more than six shots, perhaps in the belief that their innovations would secure them sales. This was rarely the case, and here I will try to explain the reasons why.
Foremost, let's explore the techniques that were used. The most simple and basic idea that was actually quite commonly deployed was that the number of shots could be increased simply by increasing the number of chambers in a cylinder. This was a technically sound plan and led to the invention of 10- or 12-shot revolvers. Parisian gunmaker Jean Chaineux used this technique for many of his own designs, including an absurd 20-shot pinfire revolver (pictured). It should be obvious why these designs weren't hugely popular with customers; they were heavy, unwieldy and hard to conceal, not to mention time-consuming to place all 20 bullets in the chambers.
Another technique, focused around keeping the cylinder at a reasonable size whilst increasing the number of chambers within it, can be seen in Dr. Alexandre Le Mat's revolvers, which employed two seperate barrels. The cylinder had an outer ring of chambers, which would be discharged from the top barrel, and an inner ring, which would be discharged from the lower barrel. Each was chambered for a different bullet. Le Mat's revolver certainly was an innovation, and it actually gained some popularity in the United States around the Civil War period. But Le Mat wasn't the only one to try this technique; Aaron C. Vaughan patented a similar design in 1862. Vaughan's design saw the two barrels placed almost side-by-side, but the cylinder system worked in much the same way. Vaughan did not, however, incorporate two different types of ammunition as Le Mat did. Vaughan's revolver wasn't very successful, probably because it was about double the thickness of a Colt percussion revolver.
There were U.S. patents in the 19th century for revolvers with multiple cylinders. One William H. Philip of Brooklyn designed a centrefire revolver with 3 cylinders, one in front of the other (pictured right). Each cylinder had 7 chambers; the first cylinder could be fully loaded with 7 bullets but the second and third cylinders could only be loaded with 6 due to the firing mechanism. A spring-loaded needle of sorts passed through the empty chambers on the second and third cylinders and when the weapon was fired, the hammer would push the needle would strike the bullets in the first chamber. Once that chamber was depleted, the needle would retract back to the second cylinder and the process would essentially be repeated until it reached the third cylinder. Philip obtained US patent #142,175 in 1873 for this revolver, and although physical examples were made, it's not likely that many were sold. Similar patents were filed by William Orr, Edward Sneider, George Gardiner and Linberg & Phillips, all of which were dual-cylinder revolvers rather than triple-cylinder models.
The so-called "pepperbox" was a type of percussion-action revolver that was popular in the mid-19th century. Pepperboxes had rotating cylinders, much like that of a regular revolver, except they did not have fixed barrels. Instead, the elongated chambers acted as the barrels. As a result, they weren't very accurate guns but they were for the most part affordable and easy to operate. An advantage of the pepperbox design was that they could have many chambers and still be of a reasonable size, whereas most conventional revolvers only ever had 5 or 6 chambers. The pepperbox design led to the advent of the short-lived "transitional" revolvers in England.
The most bizarre and probably least practical design was made by Joseph Christian Enouy of Middlesex, who patented a rotating wheel device that could be fitted on the underside of a transitional revolver (pictured). The wheel was suspended by a metal axis that ran from the grip to the barrel. Around the wheel there were eight six-chamber cylinders (48 shots in total). The idea was that once the firer had discharged all six shots from one cylinder, they could reload quickly simply by disengaging the revolver and turning the wheel so that the next cylinder would rotate in place of the previous one, and so on. Completely unsurprisingly, this design did not catch on, and it seems to have been Enouy's only venture into firearms design. I suspect it was a private venture for his own personal experimentation, as there are no records of his device ever being sold commercially.
So there you have it. Several 19th century techniques of increasing a revolver's capacity, none of which have survived the test of time. Today, most people are quite content with the 6-shot capacity of revolvers; those who want more firepower are likely to purchase an automatic pistol. That being said, I have heard of modifications being made to modern revolvers to replicate these 19th century oddities, but these are done purely out of experimentation and fun rather than practicality.