When is GROUP BY required for aggregate functions?

I have a table called myEntity as follows:

- id (PK INT NOT NULL)
- account_id (FK INT NOT NULL)
- key (INT NOT NULL.  UNIQUE for given account_id)
- name (VARCHAR NOT NULL.  UNIQUE FOR given account_id)

I don't wish to expose the primary key id to the user, and added key for this purpose. key kind of acts as an auto-increment column for a given accounts_id which will need to be manually done by the application. I first planned on making the primary key composite id-account_id, however, the table is joined to other tables, and before I knew it, I had four columns in a table which could have been one. While account_id-name does the same as account_id-key, key is smaller and will minimize network traffic when a client requests multiple records. Yes, I know it isn't properly normalized, and while not my direct question, would appreciate any constructive criticism comments.

Sorry for the rambling... When is GROUP BY required for an aggregate function? For instance, what about the following? http://stackoverflow.com/a/1547128/1032531 doesn't show one. Is it needed?

SELECT COALESCE(MAX(key),0)+1 FROM myEntity WHERE accounts_id=123;