I'm s software engineer currently mostly working with the ServiceNow platform and the Configuration Management Database and discovery processes used to populate our infrastructure data into the CMDB.
I have two main interests in graph databases. First it seems like the ideal place to store CMDB data, which is usually a collection of your infrastructure items and related applications and how they all relate to each other. This is crucial information when assessing who is impacted by outages and changes to your infrastructure. Querying a CMDB housed in a relational database turns out to be very difficult to do effectively and efficiently. It's fine if you're just looking at, say, a list of servers, etc., but as soon as you want to take relationships into account, it all breaks down.
Second, is more personal. I plan to use neo4j in a personal task/life tracking app. Everything I've seen so far for tracking tasks, etc., don't work well for how I think. I see everything as an inter-connected web of tasks, projects, related notes, etc. That is, when I visualize my life, I see it in my head like a graph - not a list. A graph database feels like a better fit to match my mental model, so I'm going to give it a try.