An oscillator is an amplifier with positive feeback, and a "frequency determining element". This can be a crystal, or an LC resonant "tank". The LC oscillator isn't usually stable enough for most applications. Here is what typically happens when you build your first LC oscillator.
First, the mechanical properties of the tuning element (usually a variable capacitor) are such that after you adjust it, it "relaxes" slightly, causing the frequency to change. So you end up adjusting it past the desired frequency, and letting it "settle" back to the target frequency.
Next, you find that as you move your hand away from the circuit, the frequency changes again. The stray capacitance added by your body affects the frequency. You end up moving your hand towards or away from the circuit in order to keep it on frequency. Next, you find that even if you hold your hand steady, the frequency is drifting and you have to move your hand again to correct for this. This drift may be from a change in temperature, voltage, or more mechanical changes.
So, the answer is "yes", but you eventually realize life might be a lot easier if you used a crystal. Crystals aren't perfect, but they are magnitudes better. They also require less power to keep the oscillations going. Good LC oscillators can be built; they will feature a well-regulated voltage supply, rigidly mounted components, and a metal shield enclosing the circuit. There will also be temperature compensation (or temperature control).
There were some remarks about frequency multiplication. This could be accomplished by introducing distortion into the oscillator, which produces harmonics. Then, another LC tank circuit is tuned to a harmonic of the original frequency. Another amplifier may be used to boost the resulting signal. This is how it was done in the days of pure analog circuitry.