Hi Dan,
Root penetration is probably more governed by the soil type. The above-ground competition with other annuals or perennials would be more a problem of competition for light I think.
I wouldn't put carrots and clover together, unless I intended to just harvest the carrot greens, which would be sort of tedious work.
I put carrots with lambsquarters this year and we harvested the lambsquarters when they were a foot high and the carrots were beginning to take over. That worked pretty well and I think the position and light was more of an issue with the root size (which was generally good but not great). These were both grown in
compost since my soil doesn't like carrots.
So maybe it depends on the 'other crop' you're growing next to the carrots.
best,
William