Seems to be an extremely common question. I put some up, please add some more ideas in the comments and I will put them up on the list!
- Wild Wadi Water Park, Jumeirah Beach Hotel
- Aquaventure Water Park, The Atlantis Hotel
- Jebel Ali Shooting Range (Pistols and Shotgun trap shooting)
- Sharjah Shooting Range (AK 47s, submachine guns etc.)
- Gold Souk (Market), Deira
- Spice Market, Deira
- Dubai Fountains, Dubai Mall, Downtown Dubai
- At the Top or At.mosphere, Burj Khalifa, Downtown Dubai
- Desert Safari (I recommend Arabian Adventures owned by Emirates)
- Brunch. For crazy I recommend Yalumba, Double Decker or Saffron. For great food I recommend Spectrum on One, Traiteur or Mina Al Qasr.
- Food, food and food: See the other Dubai Restaurant Recommendations thread
- Golf
- Trip to Abu Dhabi: Sheikh Zayed Mosque, Ferrari World,Saadiyat Island and party on Al Maya Island
- Beach at JBR
- Skydiving at the Palm/Marina
- Hot Air Balloon ride in the desert
- Camel race or horse race (winter season)
- Nightlife. Beach bars: Barasti, Nasimi, Layali. Clubs: Mahiki (my fav.), Movida, Rooftop. Casual bars: Bar Zar, Mcgettigans
- Shisha. My favourite: Iran Zamin, Dubai Marina and Mezze in Downtown Dubai. Also good is Old Palace Hotel and Pavilion in Downtown Dubai. Popular: Reem Al Bawadi.
- Ski Dubai
- Global Village (during the shopping festival)
- Trip to Oman for diving and "relaxing" (Musandam)
Please add more below!