How do I keep each PATH entry only once?

Is there a way to get rid of the duplicates without manual editing?

Preferably without installing third-party software?

If you don't mind using a PowerShell script then you can remove duplicates using the following script from the Microsoft Script Center:

Script to check for duplicate paths in PATH environment variable

Sometimes repeated installation of software can add duplicate entries into the PATH environment variable. Since environment variable has a there is a hard coded limit in the size of this variable, there are chances that you may it that limit over a period of time. This script checks the PATH environment variable and removes any duplicate path entries.

$RegKey = ([Microsoft.Win32.Registry]::LocalMachine).OpenSubKey("SYSTEM\CurrentControlSet\Control\Session Manager\Environment", $True) 
$PathValue = $RegKey.GetValue("Path", $Null, "DoNotExpandEnvironmentNames") 
Write-host "Original path :" + $PathValue  
$PathValues = $PathValue.Split(";", [System.StringSplitOptions]::RemoveEmptyEntries) 
$IsDuplicate = $False 
$NewValues = @() 
  
ForEach ($Value in $PathValues) 
{ 
    if ($NewValues -notcontains $Value) 
    { 
        $NewValues += $Value 
    } 
    else 
    { 
        $IsDuplicate = $True 
    } 
} 
  
if ($IsDuplicate) 
{ 
    $NewValue = $NewValues -join ";" 
    $RegKey.SetValue("Path", $NewValue, [Microsoft.Win32.RegistryValueKind]::ExpandString) 
    Write-Host "Duplicate PATH entry found and new PATH built removing all duplicates. New Path :" + $NewValue 
} 
else 
{ 
    Write-Host "No Duplicate PATH entries found. The PATH will remain the same." 
} 
  
$RegKey.Close() 

Source How to check for duplicate paths in PATH environment variable

You could do this with a python package called pathtub (I am the author of this package)

Installation

pip install pathtub

Usage

from pathtub import clean
# Default parameter values shown
clean(sort=True, remove_non_existent=True, remove_user_duplicates=True)

What does it do

• Removes duplicates and empty entries (;;) from the "User PATH" and "System PATH" (trailing backslash neglected when comparing two folders). Editing "System PATH" needs that python is executed with Admin rights.
• Removing from "User PATH" the entries that are in the "System PATH" (optional, enabled by default). Controlled with the remove_user_duplicates-parameter.
• Sorts PATH(s) alphabetically (optional, enabled by default). Controlled with the sort parameter.
• Removes folders from PATH(s) that do not exist on the filesystem (optional, enabled by default). Controlled with the remove_non_existent -parameter.

For more detailed description, please see the documentation in GitHub.