For every d at least 3, there exists a unimodular random graph of maximum degree d whose upper growth rate is d minus 1 and whose lower growth rate is 1, so it has no growth rate.
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A unimodular random graph with large upper growth and no growth
For every d at least 3, there exists a unimodular random graph of maximum degree d whose upper growth rate is d minus 1 and whose lower growth rate is 1, so it has no growth rate.