ClearAll[NextStep, TreePow]
NextStep[pows_List] := Module[{maxlen, sel, new, vals, knows},
  maxlen = Max[Length /@ pows[[All, "Path"]]];
  sel = Select[pows, Length[#["Path"]] == maxlen &];
  knows = pows[[All, "P"]];
  new = {};
  Do[
   vals = s["P"] + s["Path"];
   vals = DeleteCases[vals, Alternatives @@ Join[s["Path"], knows]];
   new =
    Join[
     new, <|"Path" -> Append[s["Path"], #], "P" -> #|> & /@ vals];
   ,
   {s, sel}
   ];
  new //= DeleteDuplicatesBy[#["P"] &];
  SortBy[Join[pows, new], #["P"] &]
  ]
TreePow[path_List, base_] := Module[{db, tups},
  db = <|1 -> base|>;
  Do[
   tups = Tuples[Keys[db], 2];
   tups = Select[tups, #[[2]] >= #[[1]] &];
   tups = Select[tups, Total[#] == next &];
   If[Length[tups] < 1, Abort[]];
   tups //= First;
   AssociateTo[db, Total[tups] -> (Times @@ (db /@ tups))]
   ,
   {next, Rest[path]}
   ];
  db[Last[path]]
  ]

pows = {<|"Path" -> {1}, "P" -> 1|>};
steps = Nest[NextStep, pows, 7];
LayeredGraphPlot[DirectedEdge @@@ steps[[2 ;;, "Path", -2 ;;]], VertexLabels -> Automatic]

pows = {<|"Path" -> {1}, "P" -> 1|>};
steps = Nest[NextStep, pows, 5];
assoc = Association[#["P"] -> #["Path"] & /@ steps];
Dataset[assoc]
TreePow[assoc[#], 2] & /@ Range[1, 17]

pows = {<|"Path" -> {1}, "P" -> 1|>};
steps = NestWhile[NextStep, pows, Not[MemberQ[#[[All, "P"]], 191]] &];
SelectFirst[steps, #["P"] == 191 &]["Path"];
TreePow[%, 3]

pows = {<|"Path" -> {1}, "P" -> 1|>};
steps = NestWhile[NextStep, pows, Not[MemberQ[#[[All, "P"]], 81]] &];
SelectFirst[steps, #["P"] == 81 &]["Path"];
TreePow[%, 1.1]