ROW_NUMBER Without ORDER BY

There is no need to worry about specifying constant in the ORDER BY expression. The following is quoted from the Microsoft SQL Server 2012 High-Performance T-SQL Using Window Functions written by Itzik Ben-Gan (it was available for free download from Microsoft free e-books site):

As mentioned, a window order clause is mandatory, and SQL Server doesn’t allow the ordering to be based on a constant—for example, ORDER BY NULL. But surprisingly, when passing an expression based on a subquery that returns a constant—for example, ORDER BY (SELECT NULL)—SQL Server will accept it. At the same time, the optimizer un-nests, or expands, the expression and realizes that the ordering is the same for all rows. Therefore, it removes the ordering requirement from the input data. Here’s a complete query demonstrating this technique:

SELECT actid, tranid, val,
 ROW_NUMBER() OVER(ORDER BY (SELECT NULL)) AS rownum
FROM dbo.Transactions;

Observe in the properties of the Index Scan iterator that the Ordered property is False, meaning that the iterator is not required to return the data in index key order

The above means that when you are using constant ordering is not performed. I will strongly recommend to read the book as Itzik Ben-Gan describes in depth how the window functions are working and how to optimize various of cases when they are used.

Try just order by 1. Read the error message. Then reinstate the order by (select 1). Realise that whoever wrote this has, at some point, read the error message and then decided that the right thing to do is to trick the system into not raising an error rather than realising the fundamental truth that the error was trying to alert them to.

Tables have no inherent order. If you want some form of ordering that you can rely upon, it's up to you to provide enough deterministic expression(s) to any ORDER BY clause such that each row is uniquely identified and ordered.

Anything else, including tricking the system into not emitting errors, is hoping that the system will do something sensible without using the tools provided to you to ensure that it does something sensible - a well specified ORDER BY clause.