Efficiently extendable hash function?
Use any rolling hash, e.g., the Rabin-karp rolling hash, Buzhash, or CRC.  See also the following resources:


additive hash function
Can ropes (AVL trees) be interned?
https://crypto.stackexchange.com/q/17935/351
https://crypto.stackexchange.com/q/11420/351
the answers to https://crypto.stackexchange.com/q/1558/351