Though the stakes are higher than for Project Euler, as you've pointed out, Kaggle is an excellent source of data for use in your own experiments. Many of their contests require you to be signed in to access the datasets (for legal agreements and so forth), but if you don't actually finish an entry, there's no penalty that I know of.
That being said, if you look for data sets that are specific to testing statistics procedures, like the ones at Princeton, you can test the data on different network architectures and compare it to plain regression, etc. as a benchmark.
See also here for a comprehensive list, which includes all of the Google natural language processing data.
So, Project Euler provides a great service with specific problems, but in the case of machine learning, you can use existing datasets with an architecture of your creation and compare the "answers" to conclusions that are presented online or in research papers.