Can I trade bitcoin without selling at an exchange?

Yes, it is possible to trade bitcoins directly without selling at an exchange. Many people prefer this because of their security and trust. Many exchanges were hacked in recent time, and the result of this, their bitcoins vanished without any explanation.

Another reason is its privacy. These days exchanges have similar KYC requirements as like banks needs. The KYC information is at risk of theft if the security of the exchange up to date.