First splatbook you begged your DM to approve.
I was the DM in our group, so I didn't have to beg for anything. I don't recall my players begging for any either. Most likely that was because there really weren't a lot of splatbooks for classic D&D (as opposed to AD&D 1E, which I understand had several, and 2E, which went absolutely bananas with the concept.)
My understanding is that a splatbook is a supplemental book whose purpose is to expand the options for players - races, classes, sub-classes, abilities and powers, etc. Sure, the Gazetteer series offered a few new options here and there, but I consider them more as campaign sourcebooks, since their primary purpose was to detail settings, and the new classes and sub-classes were often explicitly stated to belong to each particular place or culture. The first BECMI supplement that I think warrants the name "splatbook"
was PC1: Tall Tales of the Wee Folk, which fleshed out a variety of the
fairy and woodland creatures from the monster lists as player character
race-classes. Nobody begged, because I bought it myself, and we embraced its contents with enthusiasm - perhaps a bit too much enthusiasm, but not complete reckless abandon. I didn't allow PC treants or sidhe, and nobody complained. We had PC dryads, brownies, pixies, centaurs, pookas, hsiao, and with later entries in the series, a gremlin and a gnome as well, and it made our game a lot more fun and interesting.
That sort of splatbook I can get behind, even if the execution was a little flawed. At least it provided really different options for players. The 2E splatbooks seemed like mostly an excuse to bestow gratuitous bonuses on players for having a character concept at all. There was virtually no attempt to balance the "kits" for the various classes. When I played with the 2E group, I remember really wanting to make my fighter a peasant hero, but the kit presented in the Complete Fighter's Handbook was pitiful compared to most of the others, with the only perk being the nearly meaningless one of popularity with the lower classes of society. Compared to others that granted bonuses to hit points, attacks and damage with certain weapons, and extra proficiencies, it was pretty lame, and I would think the last thing you'd want to do would be to discourage a player from playing such a classic fantasy archetype. So, classic D&D splatbooks, great in concept, a bit shaky in execution. 2E splatbooks, terrible in concept and horribly executed.