As someone who switched from manga to anime BECAUSE I can easily see 3 episodes of all anime every cour, I never really go from one series that leads to another. Same reason why My Anime List is totally useless for me... I kinda seen it all.
Closest one that I can think of is that I first watched
Shirobako and thought it was boring as heck. It didn't help that it started with a group of girls doing cute stuff (I hate Moe girl SOL shows), so the first impression didn't help as I thought the director would be doing another Girls and Panzer. Even after the pilot episode, it was mundane stuff over and over. It was only until I was talking to my friend about how people misunderstood
Saekano from the first episode because they were watching it from the wrong perspective that he laughed at me, as I was doing the same with Shirobako.
Which was true, as Shirobako should be seen less as an anime show, and more kinda like a fictionalized documentary about how the industry works. It being relative mundane is the entire point... I mean sure, they exaggerated small things here and there, but how down to earth a lot of the mini-stories are, just leads even more credibility that the show is actually based on real life events and real live people. So upon understanding that, I blazed through the show. So I guess you can say Saekano (a show that seems to be about making Visual Novels but it isn't really) made me finish Shirobako (a show that really is about making anime). Kind of ironic actually... XD
Apart from that, I can only think of OVA's, as I've since quit trying to cover all of them as they usually are tied to an existing series or coming soon ones. Anyway, one series that comes to mind is when I was watching
Dai Mahou Toge and raving about it with a friend, and they told me to watch
Dokuro-chan. Yep same craziness all over.
(Partly to blame about my bad initial impressions about Shirobako as well, since Dai Mahou Toge, Dokuro-chan, and Shirobako has the same director. Doesn't help this same director also did Witch Craft Works as his most recent work before Shirobako, which was yet another crazy anime. Then Shirobako comes along and I'm expecting something along those lines, but after the pilot I thought I got Girls and Panzer, when what I really got was a totally different beast than anything else he has done before.
)