Short answer:
It can go on forever lit torches and repeaters
will always give a current of 15.
Long answer:
Current will travel until some of the wire is no longer loaded, if the source of the current is not inside the render distance current will not start.
(This is why Minecraft inside Minecraft will not work, there is not enough space to run all of a computer).
gaming.stackexchange.com/users/4012/Ronan
Short version: Using repeaters the travel distance is infinite.
This can be exploited to build moderately large memory buffers utilizing delay-line memory, for fun and profit.
Note that there might be practical limits to how much the game engine can handle before it blows up.
Long version: If it is not a question of distance, but rather an issue of the theoretical maximum number of wire/repeater blocks that can be powered by a single redstone torch, this number is fairly large.
When you consider the fact that redstone repeaters will reset current to 15, the actual distance itself is infinite, but there is a practical distance; loaded chunks.
This limits you to an area of 16 * 16 * 81; and if you want to keep your circuits isolated; this works out to an approximate upper limit of 11000 powered pieces of wire/repeater, unless you use nothing but repeaters and wires for turns, in which case you can almost completely tesselate all of the loaded area, allowing for a lossage of approximately 144 blocks to turning space.
Edit:
It's worth noting that in an all-repeater configuration; while you get upwards of 20000 square meters of active redstone; said redstone won't be able to power anything much; and any escape gap you make to allow devices to draw power will carry a pretty big penalty to the number of blocks that can be powered (upwards of an entire row).
Reedit:
Did a bit of a gaffe in my math:
In an all-repeater configuration, you can also use several levels; up to about 62 (allowing for bedrock and sky) layers of tightly packed redstone; giving you somewhere north of 1,200,000 meters of redstone current; or a little south of 1.2 megabits of storage in delay-line memory; 150 kilobytes.
In a single player game
this 150kB is as such the absolute maximum storage any Minecraft-based computer can have.
150kB might not sound like a lot; but if you use some sort of clock and a suitably designed buffer, it amounts to almost 10 minutes of 256-tone music, or over an hour if you limit yourself to 32 tones.
(Of course, you would have no space to build the music player and still have the memory work, but that's not the point.
;)
Threeedit:
A screenshot of a possible design of a Minecraft High-Density Memory Cell: Not pictured: A monostable circuit allowing 1-tick input and a loopback device turning it into permanent memory
This particular design uses both ^^ gates and compact repeaters; the function of the repeaters is essentially to increase isolation, increasing density and, by extension, distance; but they also double as delay-line memory.
A memory cell like this built from bedrock (4) to sky (128) over 81 chunks has an approximate maximum travel distance for a redstone pulse of 144 * 144
*
((128 - 4) / 2)
+
(128 - 4) = 1,285,756 meters; with a total capacity of 1,276,828 bits.
Note: Working out how large a stretch of redstone will have to be sacrificed in the central chunk of this memory cell in order to build a 32-tone music player; a 32-bit buffer and utilities to program the music is left as an exercise to the reader.
As is actually building it and seeing if it will actually work on the scale described.
;)
gaming.stackexchange.com/users/9401/Williham Totland
You can extend the range as far as you want in SMP, as long as other players or bots are in place to keep chunks in memory that are outside the 81 chunks that are loaded around the player.
Several of the larger redstone creations on youtube have bots stationed at regular intervals to keep the whole circuit in memory.
In single player you are limited to 81 chunks, 144 blocks .
gaming.stackexchange.com/users/3610/fredley
For those running their own server - a bit of additional information.
I'm running 1.8.
I had built a very long circuit which was not working.
I didn't count or do the math, but FYI -- I changed my view-distance parameter from the default of 10 to 15 and my circuit worked as expected.
Apparently, as I'd hoped, a longer view-distance keeps more chunks in memory.
gaming.stackexchange.com/users/13351/Pat
I've done some research into this myself.
I've got 16 sets of memory at 2bytes each.
Each memory section is approx 40 blocks long.
Redstone propagation is good up to the 5 section.
Sometimes it will get as far as the 8th section.
So current is limited to 200 blocks reliably, and up to 320 will limited reliability.
I'm on a normal render distance, and it's possible that with a higher render, you'll get farther propagation.
I'm using standard side-by-side parallel communication with a space between lines.
At 16 bits per section, that leaves 32 blocks reserved for communication.
The additional 8 is required space for routing of signals to where they need to go.
These are sustained by redstone repeaters at approx 12-14 blocks between each.
Now this is lateral communication, not vertical.
Because vertical remains within the same chunk, I believe that vertical signal propagation is infinite to the min and max of the chunk itself.
gaming.stackexchange.com/users/29312/Xero1of1
Simple: 15 Blocks is your answer.
With repeaters it can be infinite.
gaming.stackexchange.com/users/9576/Evan Darwin
If you are using mods, you could use world anchors ect to keep chunks loaded, other than that I wouldn't know whether an entity/player traveling in a minecart that is adjacent to the redstone wire would keep rendering the chunks.
gaming.stackexchange.com/users/54479/Jamie
About 5 million theoretically if you went bedrock to skylimit laying it in a spiral going up and you stood in the chunk next to your spawn chunk to double the area you have loaded.
gaming.stackexchange.com/users/108771/Andy
I know that there is a 15 block limit before a repeater is needed, however I am looking for the TOTAL distance that current can travel. Is it infinite? Can I keep adding repeaters and go on forever?  I don't mind whether "old-school" repeaters are used (2 NOT gates) or the ones added in Beta 1.5, biggest distance wins.  EDIT:  Interesting stuff guys, I would love to see some pictures of your findings!
How far can redstone current travel?
gaming.stackexchange.com
<minecraft-java-edition><minecraft-redstone>
gaming.stackexchange.com/questions/22114
gaming.stackexchange.com/users/8141/soulBit
Use repeaters and the travel distance is infinite.  Redstone current can travel infinitely with the use of repeaters.
By using repeaters redstone current can travel infinitely