It is a tough question to answer as different sources provide different information. Basically in theory, plastics could be recycled many times. However it is extremely hard to retrieve the plastics in the pure stream after use. Therefore most of the plastics are only being recycled once or downcycled into non recyclable items like fabrics. Here is a full list of plastics and what are they being recycled/downcycled into:
PET (used for water bottles, food containers, oven able film) turns into: fibres, drink bottles
PVC (used in plumbing pipes, cleaning products bottles, medical products) turns into: pipes, packaging
PP (used in condiment bottles, medicine bottles, bottle cups, tupperware) turns into: auto parts, industrial fibres, pots
HDPE (used in milk containers, wheelie bins, juice bottles) turns into: recycling bins, pallets
LDPE (used in shopping bags, food containers, gloves) turned into: plastic bags, dispensing bottles
PS (used in disposable cups, cutlery, food boxes, packaging foam) turns into: pots, tubs, trays
Miscellaneous (used in baby bottles, CDs, storage containers, number plates) turns into: car parts, pallets