I entered Korea on a 90-day visa-free entry and I want to stay longer?

The honest answer is that a visa-free short-term stay is generally not extendable just because you would like to stay longer. Visa-free entry, whether under a visa-waiver agreement or K-ETA, grants a fixed short-term visit period, and immigration treats it as a tourism or short-visit status meant to end on time rather than roll over. Extensions are granted only in genuinely exceptional situations, such as a medical emergency, a canceled flight, a family emergency, or other force majeure that makes leaving impossible, and you would apply before your permitted stay ends at your local immigration office (book through the Hi Korea site) with documents proving the emergency; even then it is discretionary and usually short.

What you generally cannot do is convert a visa-free stay into long-term life in Korea from inside the country. If your goal is to keep living, working, or studying here, you almost always need to leave and apply for the appropriate visa (work, student, working holiday, or another category) at a Korean embassy or consulate abroad, since most long-term visas must be issued from outside Korea. A few in-country status changes exist but only for specific categories and eligibility, so check your exact case. Whatever you do, do not overstay, because the fines grow with time and can trigger a re-entry ban that would hurt future plans. The safest move is to call the immigration hotline at 1345, which has English support, explain your nationality and situation, and ask specifically what is possible for you before your current stay runs out.