I didn't know that you should catch up on any fast you have missed in Ramadan before the next Ramadan arrives. Therefore I have a lot of days to catch up ad I have even lost count of the exact number of days. Can I catch these up before the next Ramadan arrives or will I have to make the payments for not completing fasts?
First, I appreciate your keenness to complete the fast, God willing Sheikh Ibn Baz says in the matter of forgetting the days of fasting following
And the meaning of his words that you have three things:
First repentance to delay fasting.
Secondly, you must fast the days of which you are certain
Thirdly, you have to feed a day for each day of delay if you can afford.
The origin of the fatwa is in Arabic here The source of fatwa and Allah know
The ruling on making up missed prayers and fasts, depends on the reason why they were missed. Here is a breakdown and the reason:
❶ - Intentional: If you miss an act of worship intentionally (meaning, you couldn't be bothered, were lazy, were ignorant, so on), then you can not make them up later. The reason for this is that each prayer (fast is similar) is for a particular and defined time (year, day). You cannot arbitrarily choose to perform them at another time because they have been commanded to occur at specific times. Allah is the One who determined when we would fast, and hence we cannot choose to do them at other times by choice (intentional). The known reference, eg, in terms of salah, is:
4:103. إن الصلاة كانت على المؤمنين كتابا موقوتا ...Verily, the prayer is enjoined on the believers at fixed hours.
In such cases, you have to only repent sincerely (with the known conditions of repentance, eg sincerity, not returning to the sin, and so on). Good deeds (eg, voluntary prayers and fasts) is also helpful in repentance.
❷ - Unintentional: If your prayer or fast was missed unintentionally, (eg, you were forced to miss it, you were genuinely tired and asleep or overslept, in battle where the Prophet once or twice prayed later due to it and cursed the disbelievers for keeping them engaged in battle till the correct time passed, etc) - then in SUCH UNINTENTIONAL circumstances, you make up the act of worship as soon as possible. (You can refer to the answer by Tarek or in other places for the penalties of late or unknown fasts/prayers etc in fatawa journals).
If this was due to period and a large amount of time passed without making up the fasts, then:
"She has to ask Allaah for forgiveness and repent to Him, and she has to fast the days that she missed and feed one poor person for each day, as was stated by a number of the companions of the Prophet (peace and blessings of Allaah be upon him). This means giving half a saa’, which is equivalent to one and a half kilograms." https://islamqa.info/en/40695
Initially I ought to mention that it is necessary to take the Qazaa (القضا) of the days which delayed its Qazaa (of Ramadhan) for a year or several years. Secondly it is necessary to point that it should pay one Mod for each day (to poor people (الفقیر)).
Of course you ought to catch them up as you said if I can catch these up before the next Ramadan arrives. But as far as I know, you must pay its Kafarah (الکفاره) as well as catching them up.
You should catch up on any fast you have missed in Ramadan before the next Ramadan arrives else you do a payments or buy a food for poor people. there are an amount for each day.
Read this to know why you should not miss any fast