The code runs as follows:

fluidPage( 
  # flexible layout function 
  h4(HTML("Think of a number:</br>Does Shiny or </br>JavaScript 
     rule?")), 
  sidebarLayout( 
    sidebarPanel( 
      # sidebar configuration 
      sliderInput("pickNumber", "Pick a number", 
      min = 1, max = 10, value = 5), 
      tags$div(id = "output") # tags$XX for holding dropdown 
    ), 
 
    mainPanel( 
      includeHTML("dropdownDepend.js"), # include JS file 
      textOutput("randomNumber"), 
      hr(), 
      textOutput("theMessage") 
    ) 
  ) 
)

The use of h4(HTML("XXX")) allows us to shrink the title a little and add some HTML line breaks (avoiding HTML escaping with the HTML function as before). tags$div(...) produces a <div> element in which to place the drop-down menu, which JavaScript will build. The mainPanel() call just contains a reference to the JavaScript file that will run on the page, a place to put the random number the server will pick, a horizontal line, and a message from JavaScript regarding whether JavaScript or Shiny rules.