Flawed proof that $\sum_{n=3}^\infty \frac{1}{n\ln{n}(\ln{(\ln{n})})}$ converges.

You've forgotten to invert the direction of inequality in your proof. It is not the case that if $a\leq b$ then $\frac1a\leq\frac1b$; in fact, (assuming all numbers involved are positive) the opposite holds: $a\leq b\implies \frac1a\geq\frac1b$. In this case, since $\ln(\ln n)\leq (\ln n)^{0.1}$, we have $\displaystyle\frac1{\ln\ln n}\geq \frac{1}{(\ln n)^{0.1}}$ and thus $\displaystyle\frac1{n(\ln n)(\ln\ln n)}\geq\frac1{n(\ln n)^{1.1}}$. But being termwise greater than the terms of a convergent series tells us nothing about the convergence of the series we're interested in.