We are collecting data to answer this question definitively. In the meantime, we have a few clues and ideas about how test scores change over time.
First, we need to distinguish between "practice" and "training." Training usually refers to doing one test to get better at other tests, or to get better at everyday cognitive tasks. To explain our take on brain training, we have another FAQ: How is this different than brain training?
Practice is less controversial. Generally, if you do one test many times, you will get better at that specific test.
Improvement can be due to getting to know the rules and controls for the test, especially the first few times you do it. You may also learn strategies that improve your score, such as a trick that combines mental steps, a repeatable method for coming to the correct conclusion, or a different way of thinking that allows you to focus better on what is important. For these reasons, your scores will improve for a while, especially relative to other people (see: Who am I being compared to in my performance?). One thing that won't help in most tests is memorization of specific puzzles; the tests are randomly generated, so they are different every time you play.
Pro tip: learn more about each test to discover the best strategies sooner. It's likely that some tests benefit more from practice than others.
Most of these sources of improvement will level off after fully learning the test and discovering all the strategies there are to discover. After that, they are more of a "pure" test of your cognitive performance. There is still a lot of room for improvement after that—such as optimizing your sleep, physical activity, and stress levels and tracking them in your journal. Physical changes may also occur in your brain to make the tests easier, like how weight training increases muscle mass; however, the science is still very much preliminary there.
We'll be following the research to better answer this question, as well as doing our own research. You can help: just keep doing the tests every day so that we can discover how your, and other users’, scores change over time.