This is one of those ideas that great minds have been chewing on for all of recorded history.

In modern thinking, we do certainly recognize different classes of intelligence. Emotional intelligence, spatial reasoning, math, language, music.

I'd say the classes of intelligence are finite. Primarily because different types of intelligence (generally) rely on distinct parts of our finite brains.

As for maximum intelligence, I like the angle taken by several SciFi authors. Generally, when an AI is elevated to the extremes of intelligence, they detach from reality and devolve into infinite naval gazing. Building simulated realities or forever lost in pondering some cosmic mystery. Humans brought to this state usually just die.

For physical, finite entities, there's almost certainly an upper limit. It's probably a lot higher than we think, though. Biological entities have hard limits on energy, nutrients, and physical size beyond which it's just impossible to maintain any organism. Infinities are rarely compatible with biology, and we will always be limited.

Even an AI is still beholden to the practicalities of moving energy and resources around. Your processor can only get so large and energy/information can only move so fast. Intelligence in AI is probably also bounded by physical constraints, same as biological entities.