For the record: I've learned about a possibility of using DAGs (in crypto) from ChuckOne, in a private conversation. In fact, I think it's one of these ideas that were "flying around": many people should have been thinking about that independently.
Could somebody provide an example of "other information for which there is no possibility of conflicts"?
I can publish "I love Mary" in one block and "I love Alice" in the other. There is no contradiction, since maybe I love both of them. But, of course, there are some possible conflicts that may arise from such a situation... 
Speaking more seriously, one may use a DAG e.g. for time-stamping purposes (since there is a well-defined partial order).
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Of course, for money transactions, there should be one and only one chain. But if we want to store some other information for which there is no possibility of conflicts similar to double-spending, we may well use another chain for that, or even a DAG.
Could somebody provide an example of "other information for which there is no possibility of conflicts"?
Speaking more seriously, one may use a DAG e.g. for time-stamping purposes (since there is a well-defined partial order).