Each router that handles a packet decrements the TTL value, until the packet reaches its destination, or TTL reaches zero, and dies.
As others have said, increasing TTL could result in packets which never die, if there is a negative cycle. Generally, if a TTL value is not large enough, the logic to try a larger TTL should probably be handled by the end-to-end clients.
If you are sure that a router is not in a cycle (tree-like topology), you could in theory safely increase the TTL value. Having said that, allowing more hops than is standard could make congestion more likely in the external network. If you have a long chain of routers between the internal and external network, as long as there is no cycle, a larger TTL value might help. Having said that, it could be quite easy for somebody to add an edge to the network and create a cycle, so starting with a larger TTL value where the packet originated in the first place is much safer.