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Nov 17, 2010
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What does “but-the-lengths-he-has-goes-to" mean?

Firstly, the phrase is incorrectly written.
It should be "but the lengths he goes to", or "but the lengths he has gone to". The first probably fits this context best.

The phrase "the lengths someone goes to" refers to the amount of effort, time, thought etc that someone expends on a some action. The suggestion is usually that that effort etc is considerable.
Since we broke up he goes to great lengths to avoid me.
A good detective will go to any lengths to find the truth.
It's surprising to what lengths some people go to follow their favourite team.

In this example the writer is expressing admiration and delight at how hard the dog works to get the teddy bear up the stairs; i.e. the lengths to which the dog goes, to achieve what it wants.