A 32 bit signed integer can hold base 10 values from −2,147,483,648 to +2,147,483,647. If they made the level counter unsigned, it would go all the way to +4,294,967,295. So my bet is either one of those numbers is the highest attainable level.
If you could play one game a second, nonstop, it would take you about 63 years to get through 2 billion levels.
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