What's an example of a function in $L^1(0,1)$ but not $L^p(0,1)$ for $p>1$? 
What's an example of a function in $L^1(0,1)$ but not $L^p(0,1)$ for $p>1$?

I've seen this answer but this is on an infinite domain. I'm interested only in $(0,1)$. I tried playing around with $\int_0^1 \frac{1}{x^a\log^b(x)}dx$ but haven't found success. I was trying to think of some transformation of the given answer to the domain $(0,1)$ but it can get messy. 
If possible, I would prefer a hint to an outright answer. 

You are definitely on the right track: I would consider something like 
$$
f = 
\begin{cases}
\frac{1}{x^a(\log x)^b} & \text{if}\ x \in (0,0.5) \\
1 &\ \text{if}\ x \in [0.5,1)
\end{cases}
$$
so that you don't need to worry about integrability at $1$. Now you only need to play around with $a$ and $b$. 
A choice that works is the following:

 $a = 1$, $b = 2$. For a proof you can check the computations for part (c) in the answer I gave to this other question.

