No, "underneath" does not imply a greater level of depth than either "beneath" or "below".
The under- indicates the subject of comparison is directly below the object or right "under its shadow". It also implies a total or partial concealment.
While the etymology of this word is slightly different (bi/by-neothan/low), it can be used interchangeably with "underneath", as @J. M. rightly pointed out in his comment on the question. However, the prefix be indicates another shade of meaning---"below the level of". Thus, if something or someone is below another, then it is not necessarily directly under but could also be at a lower level.
The White River flows through the plains beneath the majestic slopes of Mount Verde.
In this example, "beneath" cannot be interchanged with "underneath", as the plains cannot be physically under the mountain!
Check that sorry pile of newspapers. I'm sure my watch is underneath.
While it would not be incorrect to use "beneath" in place of "underneath" in this case, doing so would not be too stylistically pleasing for obvious reasons.
I'm sure my watch is underneath that sorry pile of newspapers.
Here, "beneath" can be substituted for "underneath" without inhibition!
Beneath that great city lies an unparalleled tunnel network.
"Underneath" can be used in place of "beneath" here, although "beneath" flows better, probably because of the word "lies".