Why does "SELECT POWER(10.0, 38.0);" throw an arithmetic overflow error?
Instead of meddling with Martin's answer any further, I'll add the rest of my findings regarding POWER() here.  

Hold on to your knickers.

Preamble

First, I present to you exhibit A, the MSDN documentation for POWER():


  Syntax
  
  POWER ( float_expression , y )
  
  Arguments
  
  float_expression
  Is an expression of type float or of a type that can be implicitly converted to float.
  
  Return Types
  
  Same as float_expression.


You may conclude from reading that last line that POWER()'s return type is FLOAT, but read again. float_expression is "of type float or of a type that can be implicitly converted to float". So, despite its name, float_expression may actually be a FLOAT, a DECIMAL, or an INT. Since the output of POWER() is the same as that of float_expression, it too may also be one of those types.

So we have a scalar function with return types that depend on the input. Could it be?

Observations

I present to you exhibit B, a test demonstrating that POWER() casts its output to different data types depending on its input.

SELECT 
    POWER(10, 3)             AS int
  , POWER(1000000000000, 3)  AS numeric0     -- one trillion
  , POWER(10.0, 3)           AS numeric1
  , POWER(10.12305, 3)       AS numeric5
  , POWER(1e1, 3)            AS float
INTO power_test;

EXECUTE sp_help power_test;

DROP TABLE power_test;


The relevant results are:

Column_name    Type      Length    Prec     Scale
-------------------------------------------------
int            int       4         10       0
numeric0       numeric   17        38       0
numeric1       numeric   17        38       1
numeric5       numeric   17        38       5
float          float     8         53       NULL


What appears to be happening is that POWER() casts float_expression into the smallest type that fits it, not including BIGINT. 

Therefore, SELECT POWER(10.0, 38); fails with an overflow error because 10.0 gets cast to NUMERIC(38, 1) which isn't big enough to hold the result of 1038. That's because 1038 expands to take 39 digits before the decimal, whereas NUMERIC(38, 1) can store 37 digits before the decimal plus one after it. Therefore, the maximum value NUMERIC(38, 1) can hold is 1037 - 0.1. 

Armed with this understanding I can concoct another overflow failure as follows.

SELECT POWER(1000000000, 3);    -- one billion


One billion (as opposed to the one trillion from the first example, which is cast to NUMERIC(38, 0)) is just small enough to fit in an INT. One billion raised to the third power, however, is too big for INT, hence the overflow error.

Several other functions exhibit similar behavior, where their output type is dependent on their input:


Mathematical functions: POWER(), CEILING(), FLOOR(), RADIANS(), DEGREES(), and ABS()
System functions and expressions: NULLIF(), ISNULL(), COALESCE(), IIF(), CHOOSE(), and CASE expressions
Arithmetic operators: Both SELECT 2 * @MAX_INT; and SELECT @MAX_SMALLINT + @MAX_SMALLINT;, for example, result in arithmetic overflows when the variables are of the named data type.


Conclusion

In this particular case, the solution is to use SELECT POWER(1e1, precision).... This will work for all possible precisions since 1e1 gets cast to FLOAT, which can hold ridiculously large numbers.

Since these functions are so commonplace, it's important to understand that your results may be rounded or may cause overflow errors due to their behavior. If you expect or rely on a specific data type for your output, explicitly cast the relevant input as necessary. 

So kids, now that you know this, you may go forth and prosper.