Is LocalPlayer not viable anymore?

It's most likely due to the syntax of your script. A repeat loop requires a while (condition) at the end to specify what conditions are necessary for the repeat loop to continue, and once that condition returns false, then the loop ends. Secondly, if else statements require an end at the end of the block. Here is a working version of your script:

local player = game.Players.LocalPlayer.leaderstats.Level.Value

repeat 
    wait(10)
    if player == 1 then
        wait(0.1)
        script.Parent.Text = "Private I (Lvl 1)"
    elseif player == 2 then
        wait(0.1)
        script.Parent.Text = "Private II (Lvl 2)"
    elseif player == 3 then
        wait(0.1)
        script.Parent.Text = "Private III (Lvl 3)"
    end -- Added this 'end' keyword
while (CONDITION) -- Added this statement

I do not know what is your terminal condition is, which is why I wrote "CONDITION" where you should specify what you want it to be. If you are looking for this loop to continue infinitely, then I suggest replacing repeat with while true do and removing the last line. Furthermore, make sure that this is inside a LocalScript, and that the LocalScript is inside StarterGUI. Hope this helps.

There are 2 possible problems that I can see from reading this.

1)You haven't put it in a local script which will therefore make LocalPlayer not work. 2)You forgot to put the right amount of "end" on the end of the script.

This is because local player = game.Players.LocalPlayer.leaderstats.Level.Value is saved as a number, not a property, so to fix this, define it as local player = game.Players.LocalPlayer.leaderstats.Level and change player in the if conditions to player.Value.Also, apply WaitForChild on this because the instances (Level, leaderstats) might have not loaded at the time of defining.