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The most direct competition from Microsoft would be their SQL Server Express. However that product is not as simple as it sounds. I want to call attention to another product. That is SQLLite.
SQLLite is actually a library. There is no configuration required. You link in a database engine. The contents of the database are stored in a single file.
SQLLite is implemented in C++. The API to the database is low level. This thing was made with C programmers in mind. Good luck if you want to access it through some .NET code.
You should realize that this is a scaled down database. It tries to implement most of the SQL-92 standard. But it is still low power. You don't even want to try to access the database from multiple threads. Things will not go well. I must confess that I am not a MySQL expert. So I cannot as of yet compare SQLLite with it. However I do have a personal project which is outgrowing Oracle Express Edition. MySQL may be my next bet.