In De Livera, Hyndman & Snyder (2011), a TBATS model (exponential smoothing state space model with Box-Cox transformation, ARMA errors, trend and seasonal components) for additive seasonality has been discussed. Can TBATS be used with multiplicative seasonality? Are there any restraints or numerical difficulties in TBATS or BATS with multiplicative seasonality?
The Box-Cox transformation includes logarithms, and logarithms with additive components is equivalent to multiplicative components on the original scale. There would be no point mixing Box-Cox transformations and multiplicative components.