1. What is the math involved?
You mentioned 10 weeks of time in service could be added to your pension. That's roughly .4 years. If you receive 1% per year (just guessing) and your highest year or average of whatever is in the calculation is 100k, you're looking at $400 extra/year. If you are expecting to live 40 years past retirement this is $16,000. If your avg salary used is 150k and you get 2%/year on your pension, this would be $1200/year or $48,000 over 40 years.
Put a ballpark price on your guilt or whatever costs of dealing with this situation and see if it is more or less than the added pension value. Adjust accordingly to your actual pension formula and any present/future valuation differences.
On a forum where frugality and stoicism go hand-in-hand, it cracks me up that 25 replies happened before someone looked closely at the math. Good on you, simonsez!
If you're retiring young, life expectancy is even more relevant. As I read the OP's post, all leave is subjected to the 200 hour haircut, so burning that off is pointless. Another factor is that the hours were earned gradually, at lower pay rates. When paid out, is it paid at your current, and presumably highest, rate of pay?
As to the actual math, if there's a COLA included in your pension, the amount received will continue to increase over time, like compound interest.
If you are fortunate enough to get to your target retirement date without legitimately needing these sick days, then revel in your good fortune. With a little travel hacking, you could turn $1200 (or whatever the actual number is) into a nice all expense paid trip every year.
Side note: DH has a similar plan. When he first got the job, he studied the retirement plan closely. It nets him more to stockpile his sick days until retirement, so he uses vacation days if he's sick. He works a 9/80 schedule, so he gets every other Friday off anyway, plus a generous number of paid holidays, which he is a master at combining.
A side benefit of never taking sick days is the perks associated with perfect attendance. He gets invited to annual recognition meetings, which he calls a paid half day of doing nothing + free lunch, he also regularly gets swag at the meetings. He is also considered extra badass because of his perfect health, which is pretty funny. He IS super badass, but when he does get sick, he just pulls those days out from under a different shell.
Moral of the story is that it pays to really learn your company's benefit system and work it for maximum impact on your life now and in retirement. Yes, I realize a lot of people don't stay long in their jobs these days, but for those who do, this is a great strategy to put extra bucks into your pocket.
If your job doesn't offer the perks you'd like to receive, go find another one that does, please don't whinge about it here, because it isn't relevant to the OP's question. This thread isn't as much about sick leave ethics as it is about maximizing the value of the sick time accrued. Or more precisely, learning how to wring maximum benefit from your employer's plan.