WALS Chapter 27 describes a number of functions for reduplication. All examples below come from chapter 27 unless otherwise noted.

On nouns

Pangasinan uses reduplication to mark plurals:

báley "town"
balbáley "towns"

Ilocano uses reduplication on nouns to mark a distributive plural:

sábong "flower"
sabsábong "various flowers"

Ilocano also uses reduplication to turn nouns into reciprocal verbs:

bales "revenge"
balembales "to avenge each other"

Chukchi uses reduplication to mark the absolutive case:

jokwat "eider duck"
jokwajow "eider duck (ABS)"

Kayardild uses reduplication to turn nouns into adjectives:

kandu "blood"
kandukandu "red"

Yawelmani uses reduplication to mark associatives:

k'ɔhis  "buttocks"
k'ɔk'ɔhis  "one with large buttocks"

On verbs

Luiseño uses reduplication to mark the number of an inherent object:

lawi "to make a hole"
lawlawi "to make two holes"
lawaláwi "to make more than two holes"

Mountain Arapesh uses reduplication to mark carelessness on the part of the subject or agent:

su "to touch"
susu "to touch all over"

Luiseño also uses reduplication to turn a verb into an adjective:

lepi "to tan, soften"
lepélpiš "pliable"

Ancient Greek used reduplication to mark the perfect aspect (Ancient Greek Press Book Chapter 42):

γράφ "write"
γέγραφ "have written"

Tigak uses reduplication to turn a verb into a person who regularly performs that action:

giak "send"
gigiak "messenger"

On Adjectives

Turkish uses reduplication to mark intensity:

beyaz "white"
bembeyaz "very white"

Automatic Reduplication

Additionally, automatic reduplication is possible, where an affix requires reduplication of the root with no additional meaning. Chapter 27 gives this example from Ilocano:

singpet "behave" + agin-
aginsisingpet "pretend to behave"