In structural equation modeling and confirmatory factor analysis, the most complex model is the saturated model that perfectly reproduces all observed variable means, variances, and covariances. The saturated model is "just identified", that is, it has zero degrees of freedom and by definition fits the data perfectly.

Great! thanks! Is there any criteria that would make a less complex model acceptable?

You would check the model fit (chi-square test) for non-saturated (overidentified) models. A non-significant chi-square test would indicate that the model is acceptable.

If you think of a set of data in terms of information, each measurement is one piece of information. So with N measurements, any model with N terms will fully predict the data. With a regression model and 10 pieces of data, a model with a constant plus 9 predictor variables will predict the data perfectly.

And pointlessly. Models are methods of data reduction. We choose the simplest model that gets us as close as possible to the data. Adding more terms to the model will improve fit, but we have to draw the line somewhere. So we select only those predictors that improve model fit more than you would expect by chance. Actually, t (as in t-test) has a simple interpretation : a t value of 1 means that the variable improves model fit exactly as much as you would expect by using a variable composed of random data.