おまけに・その上・しかも: the difference
おまけにN2
Meaning
In addition, On top of that, What is more, To make matters worse
Past papers
Example
家を掃除した。御負けにランチを作っといた。
I cleaned the house. On top of that, I also made lunch.
その上N2
Meaning
Besides, In addition, Furthermore
Past papers
Example
彼は勇ましい。その上優しいのだから、言うことない。
He's brave. On top of that, he's kind — I couldn't ask for more.
しかもN2
Meaning
Moreover, Furthermore
Past papers
Example
私の鼻はとても大きく、しかも曲がっている。
My nose is big and what's more, it is crooked.
How to tell them apart
「おまけに」: the strongest feeling; used for complaints and piled-up bad luck.
「その上」: works in writing and speech; neutral 'in addition'.
「しかも」: an even more striking point on top of an already striking one; carries surprise.
「それに」: the plainest, casual 'and also'.