It's not possible to do what you want with just a regular expression. A character can only be part of one match, so once you've captured 'bigFamily_' you can't capture it again as part of bigFamily_middleFamily_ within the same regular expression.
What I would do is just capture each *family part and rebuild all the possible combinations:
var = 'bigFamily_middleFamily_littleFamily_childName';
subfamilies = regexp(var, '[a-zA-Z0-9]+_', 'match');
combinations = num2cell(logical(tril(ones(numel(subfamilies)))), 2);
families = cellfun(@(c) strcat(subfamilies{c}), combinations, 'UniformOutput', false)
Or if you don't want the trailing '_':
var = 'bigFamily_middleFamily_littleFamily_childName';
subfamilies = regexp(var, '[a-zA-Z0-9]+(?=_)', 'match');
combinations = num2cell(logical(tril(ones(numel(subfamilies)))), 2);
families = cellfun(@(c) strjoin(subfamilies(c), '_'), combinations, 'UniformOutput', false)
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