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I'm surprised no other answer has mentioned this: pointers allow you to create non-contiguous and non-linear data structures, wherein an element may be related to multiple others in complex ways.

Linked lists (singly, doubly, and circularly linked), trees (red-black, AVL, trie, binary, space partitioning…), and graphs are all examples of structures that can be constructed most naturally in terms of references rather than just values.