We consider estimation of average treatment effects given observational data with high-dimensional pretreatment variables. Existing methods for this problem typically assume some form of sparsity for the regression functions. In this work, we introduce a debiased inverse propensity score weighting (DIPW) scheme for average treatment effect estimation that delivers $\sqrt{n}$-consistent estimates when the propensity score follows a sparse logistic regression model; the outcome regression functions are permitted to be arbitrarily complex. We further demonstrate how confidence intervals centred on our estimates may be constructed. Our theoretical results quantify the price to pay for permitting the regression functions to be unestimable, which shows up as an inflation of the variance of the estimator compared to the semiparametric efficient variance by a constant factor, under mild conditions. We also show that when outcome regressions can be estimated faster than a slow $1/\sqrt{ \log n}$ rate, our estimator achieves semiparametric efficiency. As our results accommodate arbitrary outcome regression functions, averages of transformed responses under each treatment may also be estimated at the $\sqrt{n}$ rate. Thus, for example, the variances of the potential outcomes may be estimated. We discuss extensions to estimating linear projections of the heterogeneous treatment effect function and explain how propensity score models with more general link functions may be handled within our framework. An R package \texttt{dipw} implementing our methodology is available on CRAN.