There are a number of integer sequences which are known to have a few "ordinary" size values, and then to suddenly grow at unbelievably fast rates. The TREE sequence is one of these sequences, which starts "1, 3" and then grows to an unimaginably large value which completely dwarfs even things like Graham's Number. Another example is given by the sequence of Ackermann numbers, which also has an extremely large third term, though not as large as that of TREE(3).
I'm interested in a variant of the above concept: integer sequences which seem to start off normally, then have one or a few values which then become mind-bogglingly large, and then which end up going back to "ordinary-sized" values for the rest of the sequence. Does anyone know of things like this which arise "naturally," perhaps in the context of graph theory or combinatorics or something similar?
Obviously one can construct sequences that fit this pattern by splicing things together, but I'm mostly interested in the case where this behavior somehow occurs in some kind of natural integer sequence.