How to break a long line with multiple bracket pairs?

Use a :

config["network"]["connection"]["client_properties"]["service"] = 
    config["network"]["connection"]["client_properties"]["service"].format(
        service=service
    )

You also could use a variable for better reading:

client_service = config["network"]["connection"]["client_properties"]["service"]
client_service = client_service.format(service=service)

# If you are using the value later in your code keeping it in an variable may
# increase readability
...
# else you can put it back
config["network"]["connection"]["client_properties"]["service"] = client_service

The brackets permit implicit line continuation. For example,

config["network"
]["connection"
]["client_properties"
]["service"] = config["network"]["connection"]["client_properties"]["service"].format(
service=service)

That said, I don’t think there’s any consensus as to which line each bracket should go on. (Personally, I’ve never found any choice that looks particularly “right”.)

A better solution would probably be to introduce a temporary variable.

d = config["network"]["connection"]["client_properties"]
d["service"] = d["service"].format(service=service)

Considering the fact that Python works with references you can do the following:

properties = config["network"]["connection"]["client_properties"]
properties["service"] = properties["service"].format(service=service)