Can we find a function $f(t)≥1$ such that the function $t^{a}/(f(t)exp(t^{2/3+b/loglog t})$ is increasing. Let $a,b$ be two positive real numbers.

Then my question is: Can we find a function $f(t)≥1$ such that the function 

$$
\frac{t^{a}}{f(t)\exp\left(t^{2/3+\frac{b}{\log\log t}}\right)}$$ is increasing for all $t≥2$.
The answer is no. Put 
$$g(t)=\frac{t^{a}}{f(t)\exp\left(t^{2/3}+\frac{b}{\log\log t}\right)}.
$$

If $f(t)\geq 1$,  then $$g(t)\leq \frac{t^{a}}{\exp\left(t^{2/3}+\frac{b}{\log\log t}\right)}.
$$
But the RHS goes to zero as $t\to \infty$, so $g(t)\to 0$, and $g$ is increasing and positive, so it must be zero. Which is nonsense.
You can pick your favorite function $g(t)$ which is both increasing and at least $1$ (say, pick $g(t) = e^t + 1$), set that expression equal to $g(t)$, and solve for $f(t)$ algebraically.