Games are designed to challenge human players, but this also makes it challenging to generate tests for games automatically. Neural networks have therefore been proposed to serve as dynamic test cases trained to reach statements in the underlying code, similar to how static test cases consisting of event sequences would do in traditional software. The NEATEST approach combines search-based software testing principles with neuroevolution to optimise such dynamic test cases. However, since NEATEST is designed as a single-objective algorithm, it may require a long time to cover even simple program states and it may get stuck trying to reach unreachable program states, which is particularly problematic as testing usually requires creating test cases for as many coverage goals as possible. In this paper, we therefore propose to treat the neuroevolution of dynamic test cases as a many-objective search problem. By targeting all coverage goals at the same time, easy goals are covered quickly, and the search can focus on more challenging ones. We extend the state-of-the-art many-objective test generation algorithms MIO and MOSA as well as the state-of-the-art many-objective neuroevolution algorithm NEWS/D to generate dynamic test cases. Experiments on a dataset of 20 SCRATCH games show that extending NEATEST to target several objectives simultaneously increases the average branch coverage from 75.88% to 81.33% while reducing the required search time by 93.28%.