How can I limit displayed nodes by term reference from current user? I am trying to create a view that to show content classified by taxonomy term and the hierarchy level of the logged in user within that taxonomy vocabulary.

For instance, the content is organized in 10 different levels of a taxonomy ranking, and each user is progressing through each level of that taxonomy (in an educational pathway), and we want to show all content of those levels that the user has reached and could have possibly have gone through up to that point.

For example if the user has reached level 5 (or has been given that rank by admin), the view will show all content for levels 1 through 5, but not anything higher.  If user is level 7, then the view will show all content with taxonomy 1 through 7 but nothing higher.

How can I do that? How can I reference the user in a view and take their taxonomy term to limit the view's content based on that?
To control access to your content based on taxonomy terms you need to use modules like Taxonomy Access Control or Permissions by Term. With proper configuration these will prevent users from viewing content they don't share terms with, even if they can guess the content links for example. 

You can create Views that lists content based on shared taxonomy terms, but Views don't control access to original nodes pages. If a user can guess the node paths, or you have content listed on Taxonomy pages and the Taxonomy pages aren't access-controled then they will be able to see the content links there and click through. 

Using one of those modules you will be able to control access through terms alone, or you may need to create a new user Role for each term, it depends on your details. 

Once you solve the Permissions problem you can create Views that would show content that the current user shares terms with. 

The easiest way to approach this is not to rely on weight/order at all. You can treat terms like badges: if a user reaches 3rd level, they would be tagged with the first 3 terms, and not only the third term. Drupal doesn't care if you name them "level-1" and "level-2" or "beginner" and "medium". You just have to make sure to add terms to the user once they pass a level, and keep all the terms they "collected" before. 

If you go for only the one highest term they have reached you have to take care of the removing the current term when a new one is reached, it introduces complexity that isn't necessary if you're relying on taxonomy terms. 

There are still ways to show only the highest term if you want to use it to rank a user. 

To create a page showing content that the current user shares terms with you use Relationships to the taxonomy reference field. The details are almost the same as in this QandA: Based on user&#39;s country filter content of same country?