Should I use MDF, plywood, or something else when building a desk?
Any sheet goods are going to be lacking in strength in terms of deflection so your construction will have to compensate for this. You'll want to create rigidity using geometry of some kind like this torsion box or maybe by adding solid wood stretchers and aprons. As far as materials go, I'd be partial to MDO for a project like this but here's your options.


Plywood is the strongest of the sheet goods and takes mechanical fasteners by far the best. But it's either a terrible surface to paint (if you get construction grade) or too nice/expensive to cover up (if you get cabinet grade). 
MDF paints well but is monstrously heavy, is the most prone to sagging, and will not take fasteners in its edges.
Particle board is inexpensive but it's flimsy and is almost impossible to seal up well enough to protect it from water damage unless you cover it with laminate but then you've just got a counter-top.
Melamine has an durable, stain proof film on both faces but as such it cannot be painted. It's also susceptible to water damage as it's just particle board in the center.
MDO is a sheet of plywood covered in a thin layer of MDf. This gives it the strength and versatility of plywood while keeping the weight down, but it also takes paint just like MDF. It's only draw back is that the edges are not finish grade in the traditional sense so they'll need to be banded or captured in some fashion, and it's a bit more pricey than MDF or CDX.