Not exactly sure what you mean by influential, since an inanimate object like a yoyo can only influence my play style or feelings to a small degree. I’ll take it to mean the most “important,” since I’m pretty sure that’s what you mean as well.
Having over 100 yoyos makes it incredibly hard to narrow this down, so I’ll keep it to a few of the most important.
General Yo Hatrick - everyone’s probably bored of this story by now, cause I always post about it, but my Hatrick was the first “high end” yoyo I ever bought. I got it directly from Ernie at Nationals in 2009, which started a long relationship of buying his latest throws at various contests from him, and strong love of General Yo throws. My best friend and I made a trip out of the contest, and we both bought our Hatricks together (and still have them) after trying out a bunch of yoyos at the contest and deciding that General Yo’s throws were by far the best. So much nostalgia associated with this thing, both due to the experiences surrounding that time, and the yoyo itself. At the time, and for awhile, it was the smoothest, best yoyo I’d ever thrown. This is one of about 3ish yoyos I’d absolutely never, ever let go of for any reason what so ever.
A-RT Grail - I love telling this story about obtaining my Grail, which was my “white whale” yoyo for the longest time.
Butter - My girlfriend got me this for our 2nd anniversary, and held the “number one favorite yoyo” spot for awhile… pretty sure it still does. I also got it right when I got back into throwing, and it showed me JUST how much better yoyos had gotten since 2013.
I definitely have more throws I “can’t live without” but these three are just on another level for me. Also, some honorable mentions: Panorama/Diorama (I currently use these the most), Free Solo and KLR (probably my all time most thrown yoyos), and Peak (nostalgia city baby).