Let's assume your actual input/output has entropy (which it doesn't for a given string or output hash value).
If I rehash
606ec6e9bd8a8ff2ad14e5fade3f264471e82251 10x times with the same SHA1 algorithm, would the entropy keep on decreasing?
Yes, but by ever so tiny amount. Hashes are designed so that every bit is dependent on all the bits of the input. Hashes do get into so called cycles after iterating quite a few times, showing that entropy can and does decrease. It is however unlikely to influence practical situations.
The initial hash will of course extract the entropy into a maximum of slightly less than 160 bits.
If so, would it ever reach a dangerous level to a point where AAA could be recovered?
That's of course nonsense statement; the amount of entropy doesn't influence the one-way-function property of a cryptographic hash function at all.
Let's take an extreme and assume that, after a while, all many messages will generate the same hash (a next to impossible situation in itself). Then the (Shannon) entropy will be lowered. So in this situation it becomes more likely to generate this iterated hash by trying out different messages.
However, if you try and revert it you will find many messages instead of one, and it is impossible to detect which one is the original input message. So if anything, you've just made the function harder to reverse.
As example: I've "hashed" all bits of a fully random message into a single bit by XOR-ing them together. Entropy is hugely decreased to a single bit worth of it. Which message did I use to create this one bit of entropy, even if you (get to) know it is