In case anyone is curious, the most aggressive form of regularization is sometimes called L0 regularization or sparse regression. Here, there is a penalty term for every additional parameter used in the model. This allows you to define a cost associated with model complexity (i.e. if an improvement to the fit is not greater than some value, the number of parameters is not increased). Solving for the best set of parameters can be tricky and thus LASSO (or L1 regularization) is often a good substitute.