Why does SQL Server "Compute Scalar" when I SELECT a persisted computed column?
Just to summarise the experimental findings in the comments this seems to be an edge case that occurs when you have two computed columns in the same table, one persisted and one not persisted and they both have the same definition. 

In the plan for the query

SELECT id5p
FROM dbo.persist_test;


The table scan on persist_test emits only the id column. The next compute scalar along multiplies that by 5 and outputs a column called id5 despite the fact that this column is not even referenced in the query. The final compute scalar along takes the value of id5 and outputs that as a column called id5p.

Using the trace flags explained in Query Optimizer Deep Dive – Part 2 (disclaimer: these trace flags are undocumented/unsupported) and looking at the query

SELECT id5,
       id5p,
       ( id * 5 )
FROM   dbo.persist_test 
OPTION (QUERYTRACEON 3604, QUERYTRACEON 8606);


Gives the output 

Tree Before Project Normalization

LogOp_Project

    LogOp_Get TBL: dbo.persist_test dbo.persist_test TableID=1717581157 TableReferenceID=0 IsRow: COL: IsBaseRow1002 

    AncOp_PrjList 

        AncOp_PrjEl QCOL: [tempdb].[dbo].[persist_test].id5

            ScaOp_Arithmetic x_aopMult

                ScaOp_Identifier QCOL: [tempdb].[dbo].[persist_test].id

                ScaOp_Const TI(int,ML=4) XVAR(int,Not Owned,Value=5)

        AncOp_PrjEl QCOL: [tempdb].[dbo].[persist_test].id5p

            ScaOp_Arithmetic x_aopMult

                ScaOp_Identifier QCOL: [tempdb].[dbo].[persist_test].id

                ScaOp_Const TI(int,ML=4) XVAR(int,Not Owned,Value=5)

        AncOp_PrjEl COL: Expr1004 

            ScaOp_Arithmetic x_aopMult

                ScaOp_Identifier QCOL: [tempdb].[dbo].[persist_test].id

                ScaOp_Const TI(int,ML=4) XVAR(int,Not Owned,Value=5)


Tree After Project Normalization

LogOp_Project

    LogOp_Get TBL: dbo.persist_test dbo.persist_test TableID=1717581157 TableReferenceID=0 IsRow: COL: IsBaseRow1002 

    AncOp_PrjList 

        AncOp_PrjEl QCOL: [tempdb].[dbo].[persist_test].id5

            ScaOp_Identifier QCOL: [tempdb].[dbo].[persist_test].id5

        AncOp_PrjEl QCOL: [tempdb].[dbo].[persist_test].id5p

            ScaOp_Identifier QCOL: [tempdb].[dbo].[persist_test].id5

        AncOp_PrjEl COL: Expr1004 

            ScaOp_Identifier QCOL: [tempdb].[dbo].[persist_test].id5


So it appears that all the computed column definitions get expanded out then during the Project Normalization stage all the identical expressions get matched back to computed columns and it just happens to match id5 in this case. i.e. it does not give any preference to the persisted column.

If the table is re-created with the following definition

CREATE TABLE dbo.persist_test (
      id            INT NOT NULL
    , id5p          AS (5 * id) PERSISTED
    , id5           AS (5 * id)
);


Then a request for either id5 or id5p will be satisfied from reading the persisted version of the data rather than doing the calculation at runtime so the matching appears to happen (at least in this case) in column order.