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Something classical ....
Patek has made some iconic watches in the past, some of which are affordable today as most collectors has moved onto larger watches. A few that comes to my mind which is within your budget.
ref :
3445 in yg or wg, automatic with 27-460 movement, classic, look for it in un-polished condition
3998 in pt, look for something in diamond dial or black dial, cool looking watch
3800 in steel or yellow gold, a fun sports piece which everybody knows
570 large size calatrava, look for it in sub seconds, more rare, also look for unpolished case
some vintage 27-460 movement watches with or without date are highly collectable, within your budget and you will find a lot of fun searching for the perfect piece.
ref 3445, 3514, 3541, 3440, etc
5107 flat bezel calatrava in automatic movement,
5711 nautilus with sapphire case back, a personal favorite of mine,
3711 in wg, only produced in one year, very rare and collectable if budget allows
enjoy your search.