Funding Scientists at the Peak Age of Discovery Act of 2014 - Amends the Public Health Service Act to require the Director of the National Institutes of Health (NIH) to ensure that the median age of first-time researchers receiving R series (research) grants is under 40 years of age by January 1, 2019, under 39 years of age by January 1, 2022, and under 38 years of age by January 1, 2025. Excludes grants for small business innovation research, small business technology transfer, or clinical trial planning.