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Maimonides codifies the rules relating to the king in detail in Hilchot Melachim. There were certainly some limitations to his authority, some examples of which follow. (All quotes are from the Touger translation, all emphases are mine.)
Hilchot Melachim 3:8
The king may only execute people by decapitation. He may also imprison offenders and have them ...
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We are allowed to "derive benefit" from unkosher animals. How does using their excrement or carcasses not fit in that leniency? Besides, anything kosher that you eat may have somehow grown using something unkosher, and we do not check for that.
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The Talmud says:
The king or the High Priest may not be members of the [Sanhedrin's] board for [deciding when to add a month to the year]: The king on account of the upkeep of the army, and the High Priest because of the cold. [Sanhedrin 18b]
Here is what it means. Before the permanent calendar, the Sanhedrin had to decide which year would be leap year (...
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