It's unlikely it will last around 100 years, however video games as a whole haven't been around that long so it's really an estimate. Games such as chess, however, have lasted for centuries and still people do not get tired of it. League of Legends and Chess are somewhat similar but also different.
League of Legends is an ever-changing game, which means that over time, it will eventually shift into a completely different game (reworks, additions and removals, etc.) with a constant genre. Chess, however, has always been and always will be the same game, yet somehow it still attracts a relatively large audience.
Chess is a game that also does not require any cognitive skills or fine-motor control of the muscles - you only have to move glorified wooden blocks over other wooden blocks, essentially. League of Legends is a much more complex game so you really have to be trained in a lot of different aspects of it (mechanical, awareness, etc.) to be remotely decent at it. This difference in playstyle ultimately decides what will succeed - with Chess having a low-complexity playstyle, it becomes easier for young children to be introduced to early, whereas in League of Legends, you have to be constantly aware to have fun.
You also have to keep in mind that over the course of 100 years, a company can switch CEO's several times due to retirement or death, and with some companies that completely changes the major product the company is selling. Take for example, Apple, the company that invented the Macintosh, the iPod, MacBook, etc.. After Steve Jobs passed, many argue that the quality of Apple products went downhill quickly. His passing meant that a new CEO had to take over, who would make decisions for the company and the likes, which changed the way the products were sold or manufactured.
While they both train the player in one way or another, Chess is simply a much better gateway into gaming than League of Legends. So will League last over 100 years? It's possible, but with how much money Marc Merill is making he may retire at one point or another, and if he doesn't, he'll spend his last days working. Either way, there will be a new CEO to take over and make decisions for the future of the company and League of Legends.
Sorry it had to end on a sour note, but that's just the way it is, and unfortunately we have to deal with it.