Highly recommended:
Formula SAE - Design and engineering competition, to build an open-wheel performance race car, in every sense. Everything *must* be optimized, it takes the whole year or more to design and build the vehicle. Vehicle design must balance 'ideal' vehicle dynamics characteristics with stiffness, weight, power, and packaging, with a mix of steel space-frame and carbon fiber monocoque chassis designs and 4-stroke motorbike engines. Rather insane power/weight ratios (e.g. 0.37 kw/kg is not unusual).
http://students.sae.org/cds/formulaseries/about.htm
Baja SAE - Off-roading competition, focused on balancing performance with durability. Heavy durability aspect, the over 80% of the vehicles will not finish the endurance, many completely roll over, and all return entirely covered in mud.
http://students.sae.org/cds/bajasae/about.htm
If you're at a school cold enough to get a lot of snow, you might have Clean Snowmobile Challenge. This is primarily a powertrain development competition. The goals here are noise, fuel economy, and emissions. IMHO, the snowmobile competition is the most personal, also the longest, at 6 days, but with only about 18 teams total there's a lot of time to talk to sponsors and event organizers directly.
http://students.sae.org/cds/snowmobile/about.htm
For anyone who says they need a totally new challenge every year, I am currently several months into designing and testing a completely new powertrain package, re-writing all of the control strategy for a newer, faster controller (80mhz PowerPC, now with 64K RAM, less than half the weight, higher timing accuracy), and working with the rules committee to allow electronic throttle control in a sane way. Last year, our (almost entirely new) team switched to a different wheel size and tire compound and completely re-designed the chassis and suspension concept. There aren't rules changes to drive innovation, but complete concept changes and improvements driven by what competitors have done, continually 'raising the bar'.