How to work with Date calculations?

Using Mr Wizard's data the complete list of away days is

away = Join @@ DateRange @@@ Reverse @ data;

The 540th most recent absence, plus 10 years is:

away[[-540]] + {10, 0, 0}
(* {2017, 9, 2} *)

This is probably not the canonical approach but I couldn't think of a way to solve this using Date functions so I converted to AbsoluteTime and worked with Interval.

I'll start with your data in a somewhat more compact format:

data = {
 {{2016, 12, 17}, {2017, 1, 1}}, {{2014, 7, 11}, {2014, 7, 19}},
 {{2014, 3,  27}, {2014, 4, 1}}, {{2013, 12, 28}, {2014, 1, 28}},
 {{2013, 10, 25}, {2013, 11, 25}}, {{2013, 4, 28}, {2013, 5, 6}},
 {{2012, 8, 5}, {2012, 8, 26}}, {{2012, 7, 2}, {2012, 7, 6}},
 {{2011, 6, 25}, {2011, 8, 15}}, {{2010, 6, 19}, {2010, 9, 18}},
 {{2009, 6, 18}, {2009, 9, 23}}, {{2009, 3, 22}, {2009, 4, 18}},
 {{2008, 6, 19}, {2008, 10, 4}}, {{2007, 6, 16}, {2007, 10, 2}},
 {{2007, 3, 24}, {2007, 4, 6}}, {{2006, 12, 16}, {2007, 1, 4}}
};

The data in absolute interval form:

ooc = Interval @@ Map[AbsoluteTime, data, {2}];

A function to compute the number of days in that interval that intersects with another date range:

days[int_, start_, end_] :=
  DayCount @@@ List @@ IntervalIntersection[int, Interval[{start, end}]] + 1 // Tr;

Interactive search:

t0 = AbsoluteTime @ Today;

Manipulate[
  {DateObject[end],
    days[ooc, DatePlus[end, {-10, "Year"}], end]},
  {end, t0, t0 + 1*^8, 24*60^2}
]

Assuming you don't leave again I think it should suffice to simply add 10 years to the 540th day in that range from the back. I.e. something like:

days = DayRange @@@
    Partition[{
      DateObject[{2017, 1, 1}], DateObject[{2016, 12, 17}],
      DateObject[{2014, 7, 19}], DateObject[{2014, 7, 11}],
      DateObject[{2014, 4, 1}], DateObject[{2014, 3, 27}],
      DateObject[{2014, 1, 28}], DateObject[{2013, 12, 28}],
      DateObject[{2013, 11, 25}], DateObject[{2013, 10, 25}],
      DateObject[{2013, 5, 6}], DateObject[{2013, 4, 28}],
      DateObject[{2012, 8, 26}], DateObject[{2012, 8, 5}],
      DateObject[{2012, 7, 6}], DateObject[{2012, 7, 2}],
      DateObject[{2011, 8, 15}], DateObject[{2011, 6, 25}],
      DateObject[{2010, 9, 18}], DateObject[{2010, 6, 19}],
      DateObject[{2009, 9, 23}], DateObject[{2009, 6, 18}],
      DateObject[{2009, 4, 18}], DateObject[{2009, 3, 22}],
      DateObject[{2008, 10, 4}], DateObject[{2008, 6, 19}],
      DateObject[{2007, 10, 2}], DateObject[{2007, 6, 16}],
      DateObject[{2007, 4, 6}], DateObject[{2007, 3, 24}],
      DateObject[{2007, 1, 4}], DateObject[{2006, 12, 16}]
      }, 2] // Apply[Join];
days // Sort // Take[#, -540] & // First@# + Quantity[10, "Years"] &

Sort orders by earliest to latest so this ought to work.

I got DateObject[{2017, 9, 2}] as the earliest date you can apply.