I only have access to Level 1 quotes (best bid and ask) for some illiquid securities. Suppose I place a buy limit order on such a security based only on the Level 1 quotes. What are the risks I could face if I have no knowledge of the rest of the order book? How vulnerable will I be to adverse quote changes that I am unable to see?
A buy limit order is an order to purchase a security at or below a specified price. That guarantees that you pay no more than your limit price.
The only downside risk that you face is that your order is filled and the security's price then drops like a rock. However, that's investment risk and it that has nothing to do with the level of quotes that you have.