What is the expiation for excessive swearing by Allah, whether in vain or by mistake, during childhood and adulthood, given that the number is unknown and there is no financial ability to offer expiation?

There is no liability upon someone for what occurred before puberty. However, after reaching the age of accountability (تكليف), there is no expiation for an oath made without intention. As for an oath made with intention (ما عقدت فيه النية), expiation is due upon breaking it, based on the Almighty's saying: "Allah will not impose blame upon you for what is unintentional in your oaths, but He will impose blame upon you for what you intended of oaths. So its expiation is the feeding of ten needy people from the average of that which you feed your [own] families or clothing them or freeing a slave. But whoever cannot find [or afford it] - then aiyam [is] three days." (Al-Ma'idah: 89). If the specific intended oaths that were broken are unknown, then caution is obligatory. And if you are unable to afford the financial expiation, then fast.