Which fingerprinting/hashing algorithms support compounding?
This falls into a general class of hash functions known as homomorphic hash functions.

Your question is not entirely clear about what definition you are using for $+$.  If you want the hash function to satisfy $h(x+y)=h(x)+h(y)$ where both instances of the $+$ operator refers to addition in a suitable group (e.g., addition of integers, xor), then such a hash is called linear.  A CRC hash is an example of a linear hash function.

It's important to distinguish concatenation from addition.  If you are looking for a hash that satisfies

$h(x||y) = h(x)+ h(y)$,

where $||$ denotes concatenation and $+$ denotes addition, then that's a different beast.  I don't know of any standard name for it.  This is a very restrictive property.  Such hash function will always necessarily have the form

$h(x) = \sum_i h(x_i),$

where $x_i$ is the $i$th character of the string $x$.  Consequently, the only way to achieve such a property is as follows: you choose some lookup table that maps from a single character to an integer; then you apply this to all the characters in the string and sum them up.  Unfortunately, such a hash is less than ideal, as a hash function, as any two strings that have the same characters in a different will yield the same hash value.  Thus if you use this as a hash function for a hash table, you might find an increased rate of collisions.  However, it would be easy to construct such a hash function.

There are other generalizations.  For instance, maybe you want a hash function with the property that

$h(x||y) = F(h(x), h(y))$

where $F$ is some associative function.  There are standard constructions of such a hash.  For instance, CRCs achieve this (though here $F$ depends on the length of $x$).

I suggest you also look at rolling hashes, the Rabin-karp rolling hash, Buzhash, AdHash, and MulHash.  See also the following resources:


https://crypto.stackexchange.com/q/17935/351
https://crypto.stackexchange.com/q/11420/351
the answers to https://crypto.stackexchange.com/q/1558/351