For time series data usually unit root tests ("ADF, DF-GLS, KPSS, PP), seasonality test(seasonal index and using seasonal dummies in your model), normality, and the other tests such as ARCH effect, GARCH effect largely depend on your technique. I f you can let me know the exact technique, I can provide you with more information.
Panel data- Hausman test, panel unit root test, endogeneity test, testing for time fixed effect, testing for random effect using BP LM test, test for cross sectional dependence using BP LM of test of independence or Pasaran CD test, heteroskedasticity, autocorrelation etc. If you are using dynamic panel data analysis there are some additional tests to be done.
The magic answer is 1. Any fewer is useless. Any more raises complicated questions about the distribution of test statistics. I admit to being provocative. The earlier responses pertain to diagnostic tests, and of course recommendations depend on context, and of course multiple diagnostic tests can be useful. My response pertains to hypothesis testing. Your question does not distinguish.
Two other thoughts: (1) Diagnostic tests often have limited power, leaving a quandary about what to do whey a diagnostic test passes or fails. (2) Most econometricians, including me, ignore my observation and employ model search procedures. Econometrics is as much craft as science.
Adding extra value to this is challenging with many interesting answers already provided. I might add that it depends on what we might consider a "test" among other things. For example, a t-test is so common that it may not come to mind. Like the other answers seem to indicate, I tend to construct "tests" for what I suspect other scholars might wish to know or question.
Your question implies that the answer can be a number, moreover a universally valid number. Like Waqas, I disagree with this implication.
My first answer is that, ideally, you should test the validity of all assumptions that your econometric analysis depends upon. In practice, this ideal, may require more time, effort, software than is available to you. My second answer is therefore that you should test the most important assumptions. If you have access to good quality econometric software then that software will offer the tests that are often thought to be important.
Perhaps this answer does not give you the practical advice you were hoping for. That may be because you have asked a question that does not have practically useful answers.
Good doctors always prescribe fewer medicines. Same thing is applicable here. If you are an expert you can even smell data. Good econometricians should not depend on so many tests. Then why we need an econometrician. Only software can conduct all tests for us.
That depends on the problem you want to analyse and on the data situation. Your question shows that you put to much weight on econometrics as a method. I strongly support Gours view. Often, one reads articles with a mass of (test) results, where I have the impression that the authors themselves cannot interpret them.
It depends on your research objectives. Econometric analysis is used to achieve research objective. Sometime only simple "t-test" is used. In this case their is no need to apply any other test. Some of the cases are as follows:
1. Time Series Analysis: Stationary analysis, OLS (multicolinearity, autocorrelation), Cointegration.
2. Advanced Time Series: Error correction model, Sensitivity analysis, Stability analysis, Causality analysis, GMM, FMOLS, DOLS, CR, Fan Chart for forecasting, ARIMA model for forecasting and others.
As few as possible because there are short solutions to long problem. When you don't know the solution you need too many tests. Good Doctors write short prescription.