The philosopher who posed the question "What can I know?" is Immanuel Kant. Kant, an 18th-century German philosopher, focused extensively on epistemology, exploring the nature and limits of human knowledge. This question is central to his critical philosophy, especially in his major work, "Critique of Pure Reason," where he investigates how knowledge is shaped by both sensory experience and the mind's structure.

While Socrates famously stated "I know that I know nothing," expressing human ignorance and the limits of knowledge, the more direct philosophical inquiry "What can I know?" as a formal epistemological question is attributed to Kant.

Thus, Kant is credited specifically with posing the question "What can I know?" in the context of philosophical investigation into knowledge.