Aggregated from this question on ResearchGate.
Connectome datasets -- A connectome (/kəˈnɛktoʊm/) is a comprehensive map of neural connections in the brain. (via Wikipedia). Can contain functional connectivity and anatomical connectivity.
Human Connectome Project : https://www.humanconnectome.org/ (Sponsored by NIH; c.f. NDA)
1000 Functional Connectomes Project : http://fcon_1000.projects.nitrc.org/fcpClassic/FcpTable.html
For Reliability and Reproducibility researches: http://fcon_1000.projects.nitrc.org/indi/CoRR/html/index.html
My Connectome Project - characterizes changes over >1 year : http://myconnectome.org/wp/
Collaborative Informatics and Neuroimaging Suite (COINS): https://coins.trendscenter.org/
Backed by: Mind Research Network, a nonprofit 501(c)3 organization Partnered with Lovelace Respiratory Research Institute
04/18/2020: 3 out of 5 data sources are down. I suggest manually browsing these sources if you so desire.
This website is developed at the Information Sciences Institute of USC. I wonder whether the collaborate with LONI (see above).
Some datasets I found interesting from OpenNEURO:
There are other non-image data sources that might be of interest:
Database for Emotion Analysis using Physiological Signals: http://www.eecs.qmul.ac.uk/mmv/datasets/deap/ (EEGs only; no MRI)
AD临床前期联盟, China : http://www.alzheimer.org.cn/app/files/index (Presents no data.)
NBDC Human Database, Japan: https://humandbs.biosciencedbc.jp/ (mainly DNA sequences obtained with illumina dye sequencing)
The National Institute of Mental Health (NIMH) Data Archive (NDA), USA: https://nda.nih.gov/
You might also be interested in related data-processing tools: