How can I find the DC gain or the constant of a given Bode diagram with no y-axis given (H(s)=K(s(s+1))/((10s+1)(s+10)(s+100)))?
Your transfer function is given by the following expression:

\$H(s)=k\frac{s(1+s)}{(1+10s)(s+10)(s+100)}\$

As this equation is not fitting a low-entropy format, you can advantageously rewrite it as

\$H(s)=\frac{k}{1000}\frac{s(1+s)}{(1+10s)(\frac{s}{10}+1)(\frac{s}{100}+1)}\$

There is a zero at the origin, hence no gain in dc. From the given graph, it seems that the magnitude flattens out at 0 dB if I understand the comment "livello 0 dB" correctly. It means that in the above expression, the leading term must equal 1 thus asking for a value of \$k\$ equal to 1000 as correctly pointed out by Pasaba Por Aqui. The below Mathcad graph confirms this number.