How To Download Any Instagram Photo On Your iPhone or other Mobile Device

Yes, there are apps that do this for you, but they tend to be filled with ads, watermark the image, or try to steal your login information. 

STEPS:

1.  Open the Instagram app and browse to the photo you want to save.
2.  Copy the link to the photo using the share buttons.
3.  Exit Instagram & open View Source or a similar app which lets you view a page's source code.
4.  Paste the share URL into the app and retrieve the page's source code.
5.  Scroll down and find where the link for the image is. Then highlight and copy it.
6.  Exit the app and open a web browser (Safari, Google Chrome, etc). Paste the image URL into the web browser.
7.  Press and hold on the image once it loads to pop up the "Save Image" dialogue. Tap "Save Image" and you are all set!

I also made a how to video detailing this same process so check it out below if you need any additional help.