What constitutes a "series?" I keep thinking of franchises that only have two or three entries, or even just one primary entry and a variety of spin-off material. Still, there are a few ideas I can confidently say apply.
I was a huge fan of Brian Jacques's Redwall while growing up, especially the book Mossflower. It was, as far as I can recall, the first fandom I got involved with, especially in regards to fanfiction; the writing side of the fandom used to be quite active in the early 2010's, though it became significantly less so as time went on. I remember a specific story called The Sword and HIs Flowers that had a profound impact on my perception of storycrafting, and may have been the beginning of my fascination with story theory. Unfortunately, the author took all of his works down at some point, and I've been unable to find a copy of it elsewhere.
I have good memories of reading the Anne of Green Gables series with my family, which took quite a few years for us to get through. It's very difficult to carry a series solely upon the depiction of everyday relationships, and this series manages it across an entire generation-and-a-half. There isn't cinematic drama or tension, but rather charming slice-of-life that is nonetheless relatable and compelling. My favorite book was Anne of Windy Poplars, though the nature of the series makes it best (though not necessary) to read in order.
Doctor Who comes to mind, but with a regrettable stipulation of distaste for the recent direction the series has taken. I didn't mind the initial reboot, and appreciated the new approach of giving a complex character arc to the titular character. Unfortunately, once that arc had run its course, the series collapsed in on itself, with initially following episodes throwing out random ideas and failing to explore them, and more recent arcs being heavily weighed down by extremely obvious agendas lacking appropriate artistic presentation. Comparatively, the classic series may not have had much depth to it, but it was still fun and engaging. Most stories took a flat-arc approach (i.e., main character is the agency of change in the world around him), and worldbuilding was accomplished slowly and naturally. But I digress. Every Doctor has a specific episode [chain] that best showcases their character, but for me the highlight of the series is The Key to Time saga and The Five Doctors.
That's already quite a bit, even just for three, so I'll leave off for now (maybe more on the 'morrow?).