As an alumni not currently working directly with a team, but being involved in some capacity since middle school, I'll try to give a more general perspective than just specific "wins" as the exciting moment.
Most exciting moment: Making lifelong friends. As cliche as it sounds, I have friends I've met through robotics over 13+ years ago and it's a mini-reunion every year at the various regionals, off-season events, etc., as so many of us have stayed involved through volunteering, mentoring, creating new teams, and so on. In the age of Facebook, it's easy to lose track of your "friends" amongst the constant influx of updates, but FIRST allows us to have a constant connection to those we met through it. Another HUGE bonus I've recently been able to appreciate much more than when I was a student is the epic networking opportunity FIRST offers. The judges at any given FIRST competition are established professionals in their fields, often time with the ability to offer internships, or possibly even jobs - if you just take the time to ask them, "so what do you do?" One of the best kept secrets about FIRST.
Most annoying moment: No alumni FIRST league!!

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Once a TechTiger, always a TechTiger.
@mabsmith
Currently a PhD student in VA, volunteering at the DC/FL Regionals and Championships.
16 years and counting with FIRST! FIRST changed my life; how's it changing yours?[/size]