What effect does sample size have on adjusted R squared values?
Adjusted r-squared is intended to be an unbiased estimate of population variance explained using the population regression equation. There are several different formulas for adjusted r-squared and there are various definitions of population variance explained (e.g., fixed versus random-x assumptions). Most commonly, statistical software will report the Ezekiel formula which makes the fixed-x assumption.
In general, as sample size increases,
So the main take-home message is that if you are interested in population variance explained, then adjusted r-squared is always a better option than r-squared. That said, as your sample size gets very large, r-squared won't be that biased (note that for models with large numbers of predictors, sample size needs to be even bigger for r-squared to approach being unbiased).
Here's a simple function in R that simulates two Gaussian variables and copies them to inflate sample size without changing their correlation. It plots adjusted $R^2$ over increasing copies with a line at $R^2$.
It generates new data every time you run it, but always takes on more or less the same shape, asymptotically approaching $R^2$ like @JeromyAnglim described. For example, with