Make the layers in the initial composition 3D layers. Then nest it in the other comp and make the layer it creates 3D, and turn on continuous rasterisation (the sun icon) for that layer. Now elements protruding beyond the edge of the nested comp will appear.
This will not create any problems if your comp is 2D, but it might create problems if you're using 3D layers in either the nested comp or the final one. The other problem is that the bounding box shown in the comp window is dynamic for a continuously rasterised layer, so it will grow to suit the size of all the visible layers.
A less kludgy way of achieving this result would be to make the nested comp big enough to fit all the action, and to add a layer to act as guide to show the size of the final frame. You could create a solid, or a shape layer with an outline or whatever you want. Unfortunately you could not use a proper guide layer, as they do not show up in nested compositions, only in the composition in which they reside. So you'd have to turn off the pseudo-guide layer before rendering.