[What helped your applications the most?]

(Math PhD): I think two things helped me the most:

1. Three strong recommenders (all in my target subfields) including senior thesis advisor. 
2. 13 grad courses (12 math, 1 CS) + 1 grad-level reading course in 3 years, both in and around my desired subfields and in breadth.

Other notes:

1. I wrote a relatively standard and boring SOP, but it clearly stated what I did before, what I want to do, who I want to do it with, and why I want to do it at [institution]
2. The resource/opportunity/having smarter peers advantage of the elite schools is significant, but nowhere near insurmountable (undergrad outside top 50 -> admitted to top 5)

Advice:

1. take as many grad courses as humanly possible
2. Do research if possible, but don't sweat not having publications (none of the 4 people admitted to a top 5 department from my school have had one)
3. don't bomb the subject GRE 
4. build relationships with faculty early
5. I don't believe emailing faculty is necessary