This list needs cleaning.
(listed in no particular order)
Kaggle Data - A growing number of datasets used in Kaggle data analysis contests and available for any other use.
Nasdaq Data Link - mainly finance related data
NHANES - longstanding and thorough survey done by CDC
SEER - Cancer data
CDC WONDER - list of mainly CDC online databases
Healthy People Website - contains among other things links to various data sources
HCUP - collection of health related databases, focusing on US wide and state-specific samples of ER and hospital visits. Not free, but not too expensive.
Clinical Study Data Request - a way to get (tedious) access to clinical trial data
EMA Clinical Data Portal - looks like a way to get access to some clinical trial data for EMA registered studies.
MIMIC - a free and open database of critical care patient visits to a Boston hospital.
Data.gov - federak government data platform.
Analyze Survey Data for Free - Step by Step Instructions to Explore Public Microdata from an Easy to Type Website
Inter-university Consortium for Political and Social Research (ICPSR) - access to various social and behavioral sciences data.