Why do FFTs have junk on the high frequency end? Suppose I go to simulate this circuit in LTSPICE:

^{simulate this circuit – Schematic created using CircuitLab}

Where the LTSPICE sine and simulation parameters are:

SINE(0 1 1K 0 0 0 1000)
.tran 1 startup

Then I ask LTSPICE to give me an FFT with no window and 1,000,000 points:

What's all the junk at the end for? I would expect only one spike at 1KHz, not an additional one at 3KHz, etc. Does this happen to all FFTs? What controls the spikes that you get after your fundamental?