Will this cause any problems when it comes to retrieving these notifications? Will I be able to retrieve the latest notifications for a specific user quickly enough (within seconds)?
Regarding the “specific user”, you could create an index for the associated field containing the user’s unique identifier (and any additional specifications e.g. date, notification messages, etc) to assist with performance. However, this is based off my interpretation of what fields the document(s) in the notification collection have.
In terms of the latest notification, I interpret this as a newly-inserted notification document. I believe when the document in question is still within your working set, it should be fast to retrieve, the total size of the collection notwithstanding. However if your goal is to never delete any document, you might be interested in taking a look at Atlas Online Archive which will automatically archive old documents, while still having the capability of querying them.
In saying so, for optimal performance you will essentially want your working set and commonly used indexes to fit in memory to prevent minimal reads from disk.