It sounds like you're talking about someone who doesn't necessarily seek out completely new experiences (which I think is why you're not happy with the other answers), but when they do try something new, they obsessively want to try every variant of that thing.
If that's right, try this:
(note that it's been pointed out that dictionary definitions like the below over-state the association with physical collections, and that it fits seeking out each of a type of experiences just as much)
a person who attempts to complete a collection or set, especially a collector who wants to collect an example of every item in a particular field:
"This recording is a must for obsessive Sinatra completists."
"I'd only recommend this movie to Hepburn completists."
From Oxford dictionaries:
An obsessive, typically indiscriminate, collector or fan of something:
"this compilation of singles and B-sides has it usefulness for completists"
"horror completists may find the film worth a sit-through"
Those examples tend to focus on things like music and films, but it can also be used for people who want to do everything in a category of activity.
For example, in mountaineering, you get "completists" who climb one mountain in a mountain range, then want to climb all of them in that range. Here's an example in use:
...But the Munros represent the granddaddy of mountain-completist challenges, tempting obsessive-compulsive outdoorsmen and women since 1891, when Sir Hugh Munro published a list of every mountain in the Scottish Highlands over 3,000 feet above sea level
Here's another example I quite like - someone's attempt to create a guide on everything to see and do in Croydon (a district of London of... mixed repute):
This is the Completists' Guide to Croydon. It is not yet complete.
[Food and drink] [Shopping] [Things to do] [etc...]