To answer your question with a sharp fact is actually difficult.. The Hubble Space Telescope has taken some wonderfully rich Deep Field images..The later of the two That spring to mind are/ is the Ultra Deep Field. Where the telescope was pointed at a known area lacking in prominent near to us stars. I seem unable to post up a link for you;
but am confident some others will jump in here, for you..or a visit to the NASA web site 'Picture of the Day' Will reveal all this to you. How far can we see ?
Over 13 billion light years away.. Some images of galaxies that were formed very early after the formation of the Universe. Very old galaxies are viewed. That we can see 13.7 billion years worth of light image.
By careful study of the band width of the electromagnetic spectrum that we can see and the infra red parts and even into the x-ray images.. and it's not just the Hubble scope that is doing all this work. Professional astronomers across the globe are working on these images every night. Radio astronomy can see a great deal of what the visible spectrum does not reveal.. To be able to 'see' a object it must first be emitting a energy detectable and in a part of the sky that lends itself to imaging.. I am involved in optical astronomy and as such I can say we often see images from outside of this Galaxy on any clear night. Thousands of light years is close. What we see is also determined by it's luminosity, brightness, magnitude. I personally struggle with anything greater than a 13 mag.. A bigger scope and longer exposure times lets you see more.. and it becomes a complex subject indeed...
The modern Lap-top or I-pad devices can get you doing astronomy from your couch that is impressive.. Just the right tools and app's and you're on. and I hope to have helped some... Mark