1. We are asked to find the minimum integer value of the function $$y = 2^{3x+5} - 1$$. 2. The function is an exponential function with base 2, which is always positive. 3. Since $$2^{3x+5} > 0$$ for all real $$x$$, the smallest value of $$y$$ occurs when $$2^{3x+5}$$ is minimized. 4. As $$x \to -\infty$$, $$3x+5 \to -\infty$$, so $$2^{3x+5} \to 0$$. 5. Therefore, the minimum value of $$y = 2^{3x+5} - 1$$ approaches $$0 - 1 = -1$$. 6. Since $$y$$ never becomes less than $$-1$$ but can get arbitrarily close, the minimum integer value $$y$$ can take is $$-1$$. Final answer: $$\boxed{-1}$$