After a few years I have actually come to realise that I didn’t fast during the holy month of Ramadhan as soon as I entered boloogh. I actually don’t remember at what age I started to fast. Now I don’t know how many days of qadha fasts I have in total. My question is: how can I compensate this? And does this also mean I can’t do mustahab fasts (for example in Rajab & Shaban) until I have compensated my qadha fasts?

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If you knew of the rules and purposely didn't fast then you have to give the qadha of the fast and pay kaffara (which is feeding 60 individual poor people per missed fast) however if you didn't know the rule at that time and were not aware then you just have to pay the qadha now and pay kaffara takhir (which is feeding one poor person 750 grams of food per fast missed).

As for calculating how many fasts you missed, just calculate how many fasts you are certain you missed so for example I'm sure I did not fast when I completed 9 years of age however I'm uncertain of the missed fasts after then, therefore I will only be required to pay the qadha of one month of Ramadan.

As for fasting mustahab, you can fast with the intention of qadha fast in the month of Rajab and Shaban and Insha'Allah get the reward of fasting in these blessed months as well.