`DendogramPlot` accepts `Axes` as an option. Despite syntax highlighting in red of `Axes` and `AxesOrigin`, `GridLines` etc. these options seem to work with `DendogramPlot`.

Inter-cluster distance in a `Cluster` object is given as the third element.

![enter image description here][1]


Several combinations of `DistanceFunction` and `Linkage` where inter-cluster distances are highlighted in red and shown as green gridlines in the dendogram plot:

    Needs["HierarchicalClustering`"]

    Grid[{{ToString@#[[1]] <> "--" <> #[[2]]}, 
      {Replace[ Agglomerate[{1, 2, 10, 4, 8},
        DistanceFunction -> #[[1]], Linkage -> #[[2]]], 
        Cluster[a_, b_, c_, d__] -> 
        Cluster[a, b, Style[c, 18, Red, Bold], d], {0, 
        Infinity}]}, {DendrogramPlot[{1, 2, 10, 4, 8},
       DistanceFunction -> #[[1]], Linkage -> #[[2]], 
       LeafLabels -> (# &), 
       GridLines -> {None, Cases[Agglomerate[{1, 2, 10, 4, 8},
           DistanceFunction -> #[[1]], Linkage -> #[[2]]], 
          Cluster[a_, b_, c_, d__] :> c, {0, Infinity}]}, 
       GridLinesStyle -> Green, ImageSize -> 500, 
       Axes -> {False, True}, AxesOrigin -> {.75, Automatic}]}}] & /@ 
     Tuples[{{Automatic, ManhattanDistance}, {"Complete",  "Centroid"}}] // Column

![enter image description here][3]

So ... vertical axis does indeed measure the inter-cluster distances for a given `DistanceFunction` and `Linkage`.




 

For various combinations of `DistanceFunction` and `Linkage` you get the following pictures:

    {#, Agglomerate[{1, 2, 10, 4, 8}, DistanceFunction -> Automatic, Linkage -> #], 
     DendrogramPlot[{1, 2, 10, 4, 8},
     DistanceFunction -> Automatic, Linkage -> #, 
     Axes -> {False, True}, AxesOrigin -> {-1, Automatic}],
     Agglomerate[{1, 2, 10, 4, 8}, DistanceFunction -> ManhattanDistance, Linkage -> #],
     DendrogramPlot[{1, 2, 10, 4, 8},
     DistanceFunction -> ManhattanDistance, Linkage -> #, 
     Axes -> {False, True}, AxesOrigin -> {-1, Automatic}]} & /@
     {"Single", "Average","Complete", "WeightedAverage", "Centroid", "Median","Ward"} // 
     Grid[Prepend[#, {"", "EuclideanDistance-Clusters", 
     "EuclideanDistance-Dendogram", "ManhattanDistance-Clusters",
     "ManhattanDistance-Dendogram"}], 
      Dividers -> All, Alignment -> Bottom] &    

![enter image description here][4]


EDIT: What I get for Frederik's example in the comments: 

    DendrogramPlot[Prime[#] & /@ Range[30], Axes -> {False, True}, 
    AxesOrigin -> {-1, Automatic}]

![enter image description here][5]


  [1]: https://i.sstatic.net/bCCRR.png
  [2]: https://i.sstatic.net/qRMQ1.png
  [3]: https://i.sstatic.net/3Ezcq.png
  [4]: https://i.sstatic.net/eTTCG.png
  [5]: https://i.sstatic.net/WA3dX.png