Dan Sikorski pointed out an interesting loop sequence within the decimal expansion of Pi.
If you search for 169, it appears at position 40. If you then search for 40, it appears at position 70. Search for 70, ... and so on.
The sequence Dan found is: 40, 70, 96, 180, 3664, 24717, 15492, 84198, 65489, 3725, 16974, 41702, 3788, 5757, 1958, 14609, 62892, 44745, 9385, 169, 40...

My remark: the position is counted from the decimal point e.g. the string "1592" is located at position 3.

16 also first starts at position 40, but it obviously never loops back to 16, because the loop that includes 40 does not include 16 (as shown above). So I think that this is proved.

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Extra credit thoughts:

There are an infinite number of numbers like 16. For instance, 1693, 16939, 169399, 1693993, etc. all start at position 40.

Starting at position 16 are 23, 238, 2384, 23946, etc.

And there are an infinite number of numbers that will never loop back to themselves starting at each of positions
1693, 16939, 169399, 1693993, etc. and at 23, 238, 2384, 23946, etc. and at etc. etc. etc.

The only other way for a number string not to loop is to have an infinite chain of numbers which never repeat. (One way this could happen would be for each number in the chain to be bigger than the one before it, but it is far more likely that one of these chains would meander back and forth while generally increasing). It is impossible to demonstrate one of these, for obvious reasons, but that doesn't mean that they don't exist. My unprovable guess is that none exists, because at each step in the chain the next number would need to be not only one which is not earlier in the chain, but also not one of the infinite number which can be reached by working backwards from 16.