Verification prevents automated systems from attempting to try to crack accounts by using successive password tries. Verification protects your account directly, and also indirectly by making it harder for other cracked accounts to be used in fraudulent transactions that you might be involved in as a buyer or seller.
In most cases, after you have gone through verification and signed in and updated your eBay cookies, it should not be necessary to continue to go through verification each time you sign in (or to sign out). After a period of a week or so of having to go through multiple verification pages to sign in, using most browsers I no longer need to unless I delete my eBay cookies or change networks.
Here are some updated recommendations to try if you are encountering sign-in problems or verification problems.
How are you signing in? Using a bookmark? If so, you may need to update it.
If you go to the ebay.com page and use the sign-in link it will send you here:
https://signin.ebay.com/ws/eBayISAPI.dll?SignIn
Almost all of the sign-in links I have seen send you to the same page, or an alternate page that looks the same:
https://signin.ebay.com/ws/eBayISAPI.dll?SignIn2
or possibly this:
https://www.ebay.com/signin/?sgn=reg&siteid=0
This older link worked better until recently, but eBay has updated things so that this link now points to the current page at the top link:
https://signin.ebay.com/ws/eBayISAPI.dll
Try each of those links to see if one of these links works better for you if you are having difficulty signing in; if so, bookmark whichever link works best for future use and please follow up to let others know, as many other users are having difficulty signing in recently.
Do not give up immediately if you get a second verification page, in some cases you may need to get through two puzzle sliders or a puzzle plus one or two obfuscated number entries to sign in.
Other things that can interfere with signing in:
Sometimes when eBay changes the sign-in process, it causes problems for some users; sometimes those problems are due to using an old browser or obsolete cookies (though not always). Some users are just annoyed by verification; while others actually get stuck in a loop that just keeps presenting new sign-in pages or new verification pages, or just will not let them sign in at all.
Make sure you are allowing the sign-in page to load completely before attempting to sign in -- users have indicated that attempting to sign in before the page has loaded can cause problems.
Make sure your browser is set to accept cookies. Try another browser or another device, or a private or incognito browser session. If you delete your cookies, you will likely have to sign in and go through strict verification again, but if you have problems with a blocked or corrupt cookie, manually deleting your eBay cookies can help. Some browsers seem to require more verification than others; try signing in and out more than once to see if it requires verification the second time around.
Make sure your browser has javascript enabled:
https://www.whatismybrowser.com/detect/is-javascript-enabled
VPNs can also cause sign-in issues, so try disabling yours if you use one and are encountering problems.
If you get a verification slider puzzle that has no background, there are three options below the slider -- close the pop-up, reload the pop-up, or try an alternative audio version that requires entering numbers that are read aloud.
If you are able to sign in, to stay signed in there is a "keep me signed in" checkbox on the sign-in page, but there is another checkbox in your account site preferences -- it might be worth checking that setting, or unchecking, saving, rechecking and saving again. That is, if you want to stay signed in and avoid having to go through verification.
If you have multiple accounts and must sign in and out to switch, consider using a dedicated browser for each account to minimize signing in and out. A private or incognito browser session is handy if you just need to check another account without signing out of your current account.
If you have successfully signed in using Google or Facebook before, try that again -- the sign in process using those may have different dependencies and may work better for you. But if you have not signed in using those before make sure you do not accidentally create a new eBay account when you attempt to sign in.
If you get really stuck, try signing in to ebay.ca or ebay.co.uk or using the eBay app on your phone -- that may allow you to do most non-listing or non-selling related activities.
