Is it possible to create a thin client that only listens for transactions and pending transactions? I looked into using a thin client but even thin clients implement SPV (Simple Payment Verification). What if you just wanted a client that only listens for transactions/pending transactions, doesn't store the blockchain, and only uses memory.
When receiving a payment, it is critical to know whether or not the transaction that makes that payment is a legitimate Bitcoin transaction (or not).

This is determined by checking if the transaction is valid according to the network's rules- and a crucial part of this is determining whether or not the transaction's inputs are valid. In order to determine the validity of the inputs, the client must have some way of knowing the current state of the network. Generally this is accomplished in two ways:


Full node: this is software that will independently download and validate every transaction and block, thus verifying what the current state of the network is. As such, any transaction can (and will) be checked for validity.
Thin/Lite/SPV client: this is software that will download the longest chain of block headers, but not the entire block. The client can thus check the validity of transaction by requesting proof of the transaction's inclusion in a block from full nodes on the network, this is accomplished via the merkle root included in the block header. This method of transaction verification requires more trust (that miners are in fact creating valid blocks), so the thin-client must trust that this verification is being performed by other nodes on the network. 


If you are not performing verification of incoming transactions in one of these ways, then you will be easily defrauded, since you will be unable to differentiate between a valid and invalid transaction, without fully trusting someone else to supply this info to you.