How popular is League of Legends?
Below is a graph for NA server activity in terms of (estimated) number of games per hour on April 14th, 2015.
As you can see, during 2AM to 8AM (keep in mind this is west coast time - PDT), there are less than 50k games starting per hour. In fact, according to my rough estimate, during these non-peak hours there are still around 500 games starting every minute on the NA server - which is very impressive considering a game of League of Legends will usually last more than 20-30 minutes and usually involves 10 players.
On the other hand, during peak periods of the day there can easily be more than 150K new games starting in an hour. This usually happens around 7PM to 10PM. For example, on April 14th (which is a Tuesday, in case you are interested), about 270K games started on the NA server on 8PM to 9PM. That's a lot of games!
There does seem to be some variations on server activity. Here's a similar but bigger plot for most of April:
It seems that there are some day-to-day variations. Since Patch 5.7 was launched on April 8th, it is not too surprising that we see some spikes of activities right after the new patch. It would also be interesting to know whether or not the server activity correlates to LCS/LCK/LPL in any shape or form. If you are interested to analyze this, I have provided the raw data here.
A note on how number of games per hour is estimated: each game on the server has a unique Match ID associated to it which is assumed to autoincrement with time. Therefore, if I can sample, say, 10000 games that started in an hour and find the smallest and the largest Match ID in this sample, then I can estimate the total number of games that started during this hour.