7. If $2^{a}+2^{b}=2^{a+b}, 2^{a}+2^{b}+2^{c}=2^{a+b+c}$, then the maximum value of $2^{c}$ is $\qquad$

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7. $\frac{4}{3}$.
From $2^{a+b}=2^{a}+2^{b} \geqslant 2 \times 2^{\frac{a+b}{2}}=2^{1+\frac{a+b}{2}}$ $\Rightarrow a+b \geqslant 1+\frac{a+b}{2} \Rightarrow a+b \geqslant 2$.
And $2^{a+b}+2^{c}=2^{a+b+c}$, so $2^{c}=\frac{2^{a+b}}{2^{a+b}-1}=1+\frac{1}{2^{a+b}-1} \leqslant 1+\frac{1}{2^{2}-1}=\frac{4}{3}$. The equality holds when $a=b=1$.
Therefore, the maximum value of $2^{c}$ is $\frac{4}{3}$.