While I have no hard numbers, my intuition tells me that genomics has made the most progress towards open science for the simple reason that most genomics data has been open since the mid-90s. This is primarily due to the efforts of the Human Genome Project and the culture of openness it engendered (which got a great boon from a) the Bermuda Principles and b) widespread opposition to efforts to patent genes, culminating in public statements by then-President Clinton).
Beyond that, I would suggest that neuroimaging has made great strides in open science (though I might be biased/ignorant because this is my field). The Human Connectome Project (HCP), which has explicitly modeled itself after the Human Genome Project, has released more than 500 individual resting-state fMRI sessions and ultimately plans to provide 1200 sessions for download (or mail order on physical media since terabytes of data is not so easy on the bandwidth). Prior to the HCP, the fMRI Data Center (fMRIDC) provided open access data sets starting in 1999. While the fMRIDC ultimately did not endure (primarily owing to technical limitations of the day; it was too expensive to store big data back when 1 GB was considered large), it paved the way for future efforts and established a precedent in the neuroimaging community. Additionally, there are efforts such as FCP-INDI (disclaimer: I'm affiliated with them) and OpenfMRI, which aggregate data from individual studies that might otherwise have gone unshared (i.e., it's not necessarily feasible for an individual lab to set up a web server for a 30 subject study, but when this scales to over 200 subjects and spans many labs it becomes worth it). A good summary of the state of open initiatives in neuroimaging can be found here and a table detailing known open datasets can be found here.
This post has been migrated from the Open Science private beta at StackExchange (A51.SE)