Upper Bound for the DFT (FFT) Coefficients of a Bounded Signal
You can obtain a bound on the magnitude of the DFT of $x[n]$ ($|x[n]|\le 1$) as follows:

$$\big|X[k]\big|=\left|\sum_{n=0}^{N-1}x[n]e^{-jkn2\pi/N}\right|\le\sum_{n=0}^{N-1}|x[n]|\le N\max_n|x[n]|=N\tag{1}$$

Note that for signals $x[n]=e^{j2\pi nl/N}$, $l\in\mathbb{Z}$, the bound is tight. However, for most other signals with $|x[n]|\le 1$ you will find that the maximum of $|X[k]|$ is substantially smaller than the bound given by $(1)$.