As explained by Michael Pilat you cannot create your own compound operators* with custom precedence. (You could conceivably write your own parser as Leonid has worked on, or attempt to coerce the Box form with CellEvaluationFunction.)
You can however use an existing operator with the desired precedence. Looking at the table Colon appears to be a good choice. The operator is entered with Esc:Esc. Example:
SetAttributes[Colon, HoldAll]
Colon[f__, x_] := Composition[f][Unevaluated@x]
ListPlot \[Colon] Flatten \[Colon] Log@N@Accumulate@# & /@ Partition[Range@300, 100]
Which appears as, and produces:
Since raw colon is already used for Pattern this may be confusing. However, if you are willing to edit your UnicodeFontMapping.tr file you can assign any symbol you like. Here I mapped \[Colon] to Klingon A:
This was done by changing the line starting with 0x2236 in UnicodeFontMapping.tr.
* Rojo demonstrated that one can create two-dimensional compound operators, meaning use of SubscriptBox, SuperscriptBox, OverscriptBox, etc. See:
