How can I get exact P values below 0.0001 for t tests or ANOVA?

Prism always reports  P to four decimal places. If you want to know P with more decimal places, you'll need to calculate it elsewhere. Excel can do the calculations, as explained below. 

First, ask Prism to report F or t to its maximum number of significant digits -- a choice in  the t test or ANOVA parameters dialog. 

Then use Excel to compute P with more decimal places.

For t tests, use the Excel function:

=TDIST(T,df, 2)

The first value is the t ratio which Prism reports. If t is negative, enter its absolute value (some older versions of Excel got confused with negative t ratios). The second value is the  number of  degrees of freedom, which Prism reports. The third value is 2 if you want a two-tail P value, and 1 if you want a one-tail P value.   

Excel 2010 introduced a new set of statistical functions, which perhaps are more accurate in some cases. To compute the P value (two-tailed) from t, you can use this newer function:

             =T.DIST.2T(t, df)

For ANOVA, use the Excel function:

=FDIST(F,DFN, DFD)

F is the F ratio Prism reports. Be sure to enter the two DF values in the correct order (numerator, then denominator). If you get them backwards, the P value can be very far from the correct value. 

Excel 2010 introduced this new form of the function, which perhaps is more accurate

            =F.DIST.RT(F,DFN, DFD)