CTE error: "Types don't match between the anchor and the recursive part"

Exactly what it says:

'name1' has a different data type to 'name' + CAST((rn+1) as varchar(255))

Try this (untested)

;with cte as
(
select 1 as rn, CAST('name1' as varchar(259)) as nm
union all
select rn+1,nm = 'name' + CAST((rn+1) as varchar(255))
from cte a where rn<10)
select * from cte

Basically, you have to ensure the length matches too. For the recursive bit, you may have to use CAST('name' AS varchar(4)) if it fails again

You need to cast both nm fields

;with cte as
(
select  1 as rn, 
        CAST('name1' AS VARCHAR(255)) as nm
union all
select  rn+1,
        nm = CAST('name' + CAST((rn+1) as varchar(255)) AS VARCHAR(255))
from cte a where rn<10)
select * from cte

For me problem was in different collation.

Only this helped me:

;WITH cte AS (
  SELECT 
    1 AS rn, 
    CAST('name1' AS NVARCHAR(4000)) COLLATE DATABASE_DEFAULT AS nm
  UNION ALL
  SELECT 
    rn + 1,
    nm = CAST('name' + CAST((rn + 1) AS NVARCHAR(255)) AS NVARCHAR(4000)) COLLATE DATABASE_DEFAULT
  FROM cte a WHERE rn < 10)
SELECT * 
FROM cte;

Hope it can help someone else.