Digits Digging a Dungeon
Pyth, 21 bytes

huXf&gt;+H@GhT@GT0G1Qm0Q


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Explanation:

                  m0Q   generate a list of zeros of length len(input)
                        This will simulate the current depths
 u               Qm0Q   reduce G, starting with G=[0,...,0], for H in input():
   f          0             find the first number T &gt;= 0, which satisfies:
    &gt;+H@GhT@GT                  H + G[T+1] &gt; G[T]
  X            G1           increase the depth at this position by one
                            update G with this result
h                       print the first element in this list