Theoretical Maximum of DFT
A simple bound is just  the number of samples. Indeed, for
$$ X_k = \sum_{n=0}^{N-1} x[n]e^{-2i\pi kn/N}$$
then obviously:

$$ |X_k| \le \sum_{n=0}^{N-1} \left|x[n]\right|\left|e^{-2i\pi kn/N}\right|$$
hence 
$$ |X_k| \le \sum_{n=0}^{N-1} \left|x[n]\right|\le N$$
if $-1\le x[n]\le 1$. The $N$ bound is
s attained for a given $k$ when you choose a complex $x[n]=\overline{e^{-2i\pi kn/N}} = e^{2i\pi kn/N}$. If your input allows a constant signal  (equal to one), $x[n]=1$, then this is attained for the zeroth frequency $X_0 =  \sum_{n=0}^{N-1} 1 = N$. As a side note, if the signal length is a power of $2$, this is quite convenient. 

For more restricted inputs, like real, zero-average signals only, there could be a tighter bound. A recent paper presented at SampTA 2017 provides DFT bounds for some random sequences (Bernoulli): Bounds on Discrete Fourier Transform of Random Mask (published version).