Here's how to do it as of September 2020 without an external website like instadp.com

0) First, make sure you are logged into an Instagram account on the browser that you want to do this.

1) Enter a username in place of "USERNAME" in this URL:

https://www.instagram.com/USERNAME/?__a=1

2) Visit the URL.

3) Copy the number after "profilePage_" at the start. This is the userID. (It is also located under graphql.user.id in the JSON.)

4) Paste the number in this URL in place of "NUMBERHERE"

https://i.instagram.com/api/v1/users/NUMBERHERE/info/

5) Visit the URL. You will most likely see {"message": "useragent mismatch", "status": "fail"}. If you do, perform the following steps:

6) If you are in Chrome, press F12 or Ctrl+Shift+I or Ctrl+Shift+J to open DevTools.

7) In the bottom draw (where you see "Console", "What's New", Network conditions"), click "Network conditions".

Under "User agent", uncheck "Select automatically".

8) Select "Custom...", and paste an Instagram mobile user agent into the box, like this one:

Mozilla/5.0 (Linux; Android 9; GM1903 Build/PKQ1.190110.001; wv) AppleWebKit/537.36 (KHTML, like Gecko) Version/4.0 Chrome/75.0.3770.143 Mobile Safari/537.36 Instagram 103.1.0.15.119 Android (28/9; 420dpi; 1080x2260; OnePlus; GM1903; OnePlus7; qcom; sv_SE; 164094539)

9) Press Ctrl+F, and search for "hd_profile_pic_url_info".

10) Copy the URL between "url" and "width". It's a long one, so look carefully. (It is located under user.hd_profile_pic_url_info.url in the JSON.)

11) Paste the URL in the address bar, but don't press Enter. Look for all instances of "\u0026" in the URL. There should be about 3 of them, in the second half of the URL. These are the JavaScript codes for "&", but they won't work in an address bar URL (you will see Bad URL timestamp if you press Enter). Replace them all with "&".

12) Visit the URL.

Now you have your full-resolution profile picture.