What computations can NOT be performed in standard SQL? I know loops cannot be expressed in standard SQL. What other forms of processing can it not perform?
loops cannot be expressed in standard SQL


That is not the case. The relevant part of the Standard is known as SQL/PSM (Persistent Stored Modules). Its procedural paradigm includes 'loops'.

However, consider that core Standard SQL has been relationally complete (by Codd's definition of that term) since at least SQL-92, allowing for 'computations' of arbitrary complexity. Also that a SQL query is more like a specification than an implementation i.e. states the intent for the DBMS to carry out as it sees fit which may itself involve 'looping'.