You're off-by-one; the deadline to file for refund of TY2010 was Apr. 15 2014.
For the remaining (in-statute) refund years, I'm not so sure as @littleadv it's actually prohibited, but it certainly can't go through the normal logic.
I had a case involving several back years "offsetting" (but not just late filing like yours) and found that IRS systems and procedures mostly handle each year separately (they call it a "module").
Thus if it is possible to do this for your case, it probably won't be easy. I suggest you try the Taxpayer Advocate Service http://www.irs.gov/Advocate .
This is an omsbudsman function funded within the IRS but reporting independently to Congress. They have discretion which cases they take, so they won't necessarily take yours, but if they do they can apply insider knowledge to help you (though they still have to execute through line functions).
Not that it helps now, but I think it might have worked to file out of order.
I did experience one exception to the "each year separate" rule; when appeals finally awarded me a refund for one back year, the IRS gladly applied it as of that back year to a debt their systems believed (wrongly!) I owed on another year.
If you had filed 2013 and let it be processed, then filed 2011 and 2012 and those processed okay, I am quite certain they would have "diverted" those refunds to your 2013 module -- and maybe "as of" 2012 and 2013 respectively.