You don't need to start following someone to see that person's profile and recent photos and videos. As long as the user in question has a public feed, you can view it by tapping on the username. Username links appear next to comments, in search results, in mentions and other areas of Instagram.
You can search for other users in the Explore section of the Instagram mobile app. Tap the magnifying glass icon to look for usernames and tap on any result to see the relevant user's feed. If the feed has been made private, recent photos and videos cannot be accessed until you are approved as a follower. You can also tap through to user profiles when you see a username link in a comment or in your activity stream.
On the Web
Every Instagram user has a Web profile at instagram.com/username; this is another way of viewing someone's feed without following the person. As on the mobile app, you can click through on a username in a mention or comment, though the Web interface doesn't offer the ability to search for users. If the user account is protected, you'll see a padlock symbol instead of a photo and video gallery. You can also click through to a user's Instagram profile from various other locations, such as photos embedded on a website or linked to on Twitter.
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