By increasing the size of the training set the model memorize more data. Thus, will using leave one out increase the chance of overfitting?
ML model will start to "memorize" data with increasing complexity of your algorithm (and its parameters), not the size of training set.
Cross-validation is used to estimate your model performance on data that was not used to train. If you use LOOCV (k=n) then your k models will be (almost) identical. This gives you a high variance in your model evaluation and low bias regarding the final model trained on the entire data set.
Use 10-times 10-fold stratified CV if you unsure about a good value for k.