I found this cool map today. It shows the world very nearly without distortions of size, but still in a rectangle and still useful for navigation. e.g. africa is bigger than north america - as it really is, and so on.
Here is the main site in Japan and the '2' at the top of the page will show you the 5 transforms he did to achieve the result. He wants to make it more fine-grained with even less distortion.
AuthaGraph オーサグラフ 世界地図