if you are not really exceptional DON'T apply. each of the departments in top tier universities have couple of hundred applicants (in some departments maybe thousands) and they look for everything in your application. your background and and GPA, letters from well KNOWN people that they know and reputation of your undergrad school are also very important. if you do your undergrad at Stanford and then apply to MIT it's much easier to get into than if you come from a place which is not reputed.
good luck
btw, you shouldn't expect to have lots of fun by studying in these nerd schools. doing phd even in a middle ranked university is so challenging and laborious
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