I encountred this error while trying to start a full crawl on the search service application of a SharePoint 2013 farm.

Access is denied. Verify that either the Default Content Access Account has access to this repository, or add a crawl rule to crawl this repository. If the repository being crawled is a SharePoint repository, verify that the account you are using has "Full Read" permissions on the SharePoint Web Application being crawled.

After spending a lot of time googling this, I ended up founding THE SOLUTION.
Usualy this error is related to some URLs from which the search system start crawling. You can edit these URLs from the Search Service Application / Content Souces / Start Addresses

Anyway, this how you can garantee a working crawl for your Search Service ;

1. Check your Search Account

First go to you Sharepoint central admin

Click on "Manage service application", then "Search Service Application".

On this screen, you have you search account in front of "Default content access account"

This account will need some special roles and rights (permissions) to get through your data

2. Get SharePoint Web Applications being crawled

Click on "Manage service application", then "Search Service Application". On the left menu, click "Content Sources" from the Crawling section.

Here you will find at least on content source created by SharePoint for you : "Local SharePoint sites"

Edit this content souce, and notice all the "Start Addresses". You will need to configure every web application corresponding to these URLs.

Here, usualy problems came from sps3://[my-site-url]. You will see further on this ticket.

3. Giving permissions to the Search Account

Central Administration > Application Management > Manage Web Application

Here you have all your web applications, select everyone of those, just highlighting its line, and do the following :

Click on "User Policy" from the ribbon

Click "Add Users" from the popup showing, and select your Search Account. You should give Full Read permission.

4. What about the User Profile Application Service ?

Steps 1, 2 and 3 could be good enough. But what if not ?

Go to the UPS, the User Profile Service. Just the same way you went to the Search Service Application ? No, but almost, from the "Manage service application", Highlight UPS and click "Administrators" from the ribbon.

Add your Search Account and check "Retrieve People Data for Search Crawlers". You could even check "Full Control", but the first checkbox should be sufficient.

5. The Search Account itself