'ou me' strengthens a negation, Friberg's lexicon gives the examples of 'never, in no way, under no circumstances, certainly not'.
This are often translated in the English, but not always. I've put some examples I've come across in my reading recently:
ESV John 4:48 So Jesus said to him, "Unless you see signs and wonders you will not believe."
In the Greek it says 'ou me pisteusete' meaning 'you will never believe'
Ok, not wildly encouraging, but then there's:
ESV John 6:35 Jesus said to them, "I am the bread of life; whoever comes to me shall not hunger, and whoever believes in me shall never thirst.
In the Greek it says 'ou me peinase' = 'will never go hungry'
and 'ou me dipsesei' = "will never thirst".
What a comfort to know this!
ESV John 6:37 All that the Father gives me will come to me, and whoever comes to me I will never cast out.
which reads 'ou me ekbalo eko' = 'I will never throw out'
Jesus is emphatic about this, no way, under no circumstances will he ever throw us out!
ESV John 8:12 Again Jesus spoke to them, saying, "I am the light of the world. Whoever follows me will not walk in darkness, but will have the light of life."
'ou me periptese en te skotia' = "will never walk in darkness"
ESV John 8:51 Truly, truly, I say to you, if anyone keeps my word, he will never see death."
'thanaton ou me theorese eis ton aiwna' = 'he will never see death forever'
As you read John's gospel through you start to feel the culmulative effect of these ou me's!