Demonstration of full-height layout in Shiny. The secret is "position: absolute" and top/left/right/bottom set to 0. We'll add a layout like this to Shiny, but I want it to be way more customizable/nicer than the implementation you see here, so don't get too attached to this code :)

server.R

shinyServer(function(input, output) {
  output$plot <- reactivePlot(function() {
    plot(cars)
  })
})

ui.R

horizontalSplitLayout <- function(left, right, lwidth='370px') {
  bootstrapPage(
    tags$head(tags$style(type="text/css", "html, body {width: 100%; height: 100%; overflow: hidden}")),
    tags$div(style="position: absolute; top: 0; left: 0; right: 0; bottom: 0",
      tags$div(style=paste("position: absolute; top: 0; left: 0; width:", lwidth, "; bottom: 0; overflow: auto"), left),
      tags$div(style=paste("position: absolute; top: 0; left:", lwidth, "; right: 0; bottom: 0; overflow: auto"), right)
    )
  )
}

sidebarSplitLayout <- function(controls, main) {
  horizontalSplitLayout(
    left = tags$div(class="well", style="margin: 30px",
      controls
    ),
    right = main,
    lwidth = '370px'
  )
}

shinyUI(sidebarSplitLayout(
  tagList(
    textInput("foo", "Foo"),
    textInput("bar", "Bar"),
    textInput("baz", "Baz")
  ),
  plotOutput("plot", width="100%", height="100%")
))

Also see this link for an in-depth writeup of this technique.