I have had not above two _____ sickness since my arrival in this country.
You'd say the -s in the singular, so it should be -s'.
I disagree. Don't we only add an apostraphe if it's possesive, or if it's a contraction for "week is", and not for the plural?
It is possessive AND plural, hence the apostrophe after the 's'.
It is correct the plural only:... two weeks sickness...
Saxon genitive of two weeks is two weeks'